On Some Quadratic Algebras I 1/2: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials

We study some combinatorial and algebraic properties of certain quadratic algebras related with dynamical classical and classical Yang-Baxter equations.

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Veröffentlicht in:	Symmetry, Integrability and Geometry: Methods and Applications
Datum:	2016
1. Verfasser:	Kirillov, A.N.
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spelling	Kirillov, A.N. 2019-02-14T18:13:23Z 2019-02-14T18:13:23Z 2016 On Some Quadratic Algebras I 1/2: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials / A.N. Kirillov // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 142 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 14N15; 53D45; 16W30 DOI:10.3842/SIGMA.2016.002 https://nasplib.isofts.kiev.ua/handle/123456789/147419 We study some combinatorial and algebraic properties of certain quadratic algebras related with dynamical classical and classical Yang-Baxter equations. I would like to express my deepest thanks to Professor Toshiaki Maeno for many years fruitful collaboration. I’m also grateful to Professors Yu. Bazlov, I. Burban, B. Feigin, S. Fomin, A. Isaev, M. Ishikawa, M. Noumi, B. Shapiro and Dr. Evgeny Smirnov for fruitful discussions on dif ferent stages of writing [72]. My special thanks are to Professor Anders Buch for sending me the programs for computation of the β-Grothendieck and double β-Grothendieck polynomials. Many results and examples in the present paper have been checked by using these programs, and Professor Ole Warnaar (University of Queenslad) for a warm hospitality and a kind interest and fruitful discussions of some results from [72] concerning hypergeometric functions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On Some Quadratic Algebras I 1/2: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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title	On Some Quadratic Algebras I 1/2: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials
spellingShingle	On Some Quadratic Algebras I 1/2: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials Kirillov, A.N.
title_short	On Some Quadratic Algebras I 1/2: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials
title_full	On Some Quadratic Algebras I 1/2: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials
title_fullStr	On Some Quadratic Algebras I 1/2: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials
title_full_unstemmed	On Some Quadratic Algebras I 1/2: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials
title_sort	on some quadratic algebras i 1/2: combinatorics of dunkl and gaudin elements, schubert, grothendieck, fuss-catalan, universal tutte and reduced polynomials
author	Kirillov, A.N.
author_facet	Kirillov, A.N.
publishDate	2016
language	English
container_title	Symmetry, Integrability and Geometry: Methods and Applications
publisher	Інститут математики НАН України
format	Article
description	We study some combinatorial and algebraic properties of certain quadratic algebras related with dynamical classical and classical Yang-Baxter equations.
issn	1815-0659
url	https://nasplib.isofts.kiev.ua/handle/123456789/147419
citation_txt	On Some Quadratic Algebras I 1/2: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials / A.N. Kirillov // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 142 назв. — англ.
work_keys_str_mv	AT kirillovan onsomequadraticalgebrasi12combinatoricsofdunklandgaudinelementsschubertgrothendieckfusscatalanuniversaltutteandreducedpolynomials
first_indexed	2025-11-24T02:39:40Z
last_indexed	2025-11-24T02:39:40Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 002, 172 pages On Some Quadratic Algebras I 1 2 : Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss–Catalan, Universal Tutte and Reduced Polynomials Anatol N. KIRILLOV †‡§ † Research Institute of Mathematical Sciences (RIMS), Kyoto, Sakyo-ku 606-8502, Japan E-mail: kirillov@kurims.kyoto-u.ac.jp URL: http://www.kurims.kyoto-u.ac.jp/~kirillov/ ‡ The Kavli Institute for the Physics and Mathematics of the Universe (IPMU), 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan § Department of Mathematics, National Research University Higher School of Economics, 7 Vavilova Str., 117312, Moscow, Russia Received March 23, 2015, in final form December 27, 2015; Published online January 05, 2016 http://dx.doi.org/10.3842/SIGMA.2016.002 Abstract. We study some combinatorial and algebraic properties of certain quadratic algebras related with dynamical classical and classical Yang–Baxter equations. Key words: braid and Yang–Baxter groups; classical and dynamical Yang–Baxter relations; classical Yang–Baxter, Kohno–Drinfeld and 3-term relations algebras; Dunkl, Gaudin and Jucys–Murphy elements; small quantum cohomology and K-theory of flag varieties; Pieri rules; Schubert, Grothendieck, Schröder, Ehrhart, Chromatic, Tutte and Betti polynomials; reduced polynomials; Chan–Robbins–Yuen polytope; k-dissections of a convex (n+ k + 1)- gon, Lagrange inversion formula and Richardson permutations; multiparameter deforma- tions of Fuss–Catalan and Schröder polynomials; Motzkin, Riordan, Fine, poly-Bernoulli and Stirling numbers; Euler numbers and Brauer algebras; VSASM and CSTCPP; Birman– Ko–Lee monoid; Kronecker elliptic sigma functions 2010 Mathematics Subject Classification: 14N15; 53D45; 16W30 To the memory of Alain Lascoux 1944–2013, the great Mathematician, from whom I have learned a lot about the Schubert and Grothendieck polynomials. Contents 1 Introduction 6 2 Dunkl elements 18 2.1 Some representations of the algebra 6DTn . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.1.1 Dynamical Dunkl elements and equivariant quantum cohomology . . . . . . . . . . 19 2.1.2 Step functions and the Dunkl–Uglov representations of the degenerate affine Hecke algebras [138] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.1.3 Extended Kohno–Drinfeld algebra and Yangian Dunkl–Gaudin elements . . . . . . 26 2.2 “Compatible” Dunkl elements, Manin matrices and algebras related with weighted complete graphs rKn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3 Miscellany . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3.1 Non-unitary dynamical classical Yang–Baxter algebra DCYBn . . . . . . . . . . . 29 2.3.2 Dunkl and Knizhnik–Zamolodchikov elements . . . . . . . . . . . . . . . . . . . . . 31 mailto:kirillov@kurims.kyoto-u.ac.jp http://www.kurims.kyoto-u.ac.jp/~kirillov/ http://dx.doi.org/10.3842/SIGMA.2016.002 2 A.N. Kirillov 2.3.3 Dunkl and Gaudin operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3.4 Representation of the algebra 3Tn on the free algebra Z〈t1, . . . , tn〉 . . . . . . . . . 33 2.3.5 Kernel of Bruhat representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.3.6 The Fulton universal ring [47], multiparameter quantum cohomology of flag varieties [45] and the full Kostant–Toda lattice [29, 80] . . . . . . . . . . . . . . . . 36 3 Algebra 3HTn 38 3.1 Modified three term relations algebra 3MTn(β, ψ) . . . . . . . . . . . . . . . . . . . . . . 40 3.1.1 Equivariant modified three term relations algebra . . . . . . . . . . . . . . . . . . 42 3.2 Multiplicative Dunkl elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3 Truncated Gaudin operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.4 Shifted Dunkl elements di and Di . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4 Algebra 3T (0) n (Γ) and Tutte polynomial of graphs 52 4.1 Graph and nil-graph subalgebras, and partial flag varieties . . . . . . . . . . . . . . . . . 52 4.1.1 Nil-Coxeter and affine nil-Coxeter subalgebras in 3T (0) n . . . . . . . . . . . . . . . 52 4.1.2 Parabolic 3-term relations algebras and partial flag varieties . . . . . . . . . . . . . 54 4.1.3 Universal Tutte polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.1.4 Quasi-classical and associative classical Yang–Baxter algebras of type Bn . . . . . 69 4.2 Super analogue of 6-term relations and classical Yang–Baxter algebras . . . . . . . . . . . 71 4.2.1 Six term relations algebra 6Tn, its quadratic dual (6Tn)!, and algebra 6HTn . . . . 71 4.2.2 Algebras 6T (0) n and 6TF n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.2.3 Hilbert series of algebras CYBn and 6Tn . . . . . . . . . . . . . . . . . . . . . . . . 76 4.2.4 Super analogue of 6-term relations algebra . . . . . . . . . . . . . . . . . . . . . . . 79 4.3 Four term relations algebras / Kohno–Drinfeld algebras . . . . . . . . . . . . . . . . . . . 80 4.3.1 Kohno–Drinfeld algebra 4Tn and that CYBn . . . . . . . . . . . . . . . . . . . . . 80 4.3.2 Nonsymmetric Kohno–Drinfeld algebra 4NTn, and McCool algebras PΣn and PΣ+ n 83 4.3.3 Algebras 4TTn and 4STn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.4 Subalgebra generated by Jucys–Murphy elements in 4T 0 n . . . . . . . . . . . . . . . . . . . 86 4.5 Nonlocal Kohno–Drinfeld algebra NL4Tn . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.5.1 On relations among JM-elements in Hecke algebras . . . . . . . . . . . . . . . . . . 89 4.6 Extended nil-three term relations algebra and DAHA, cf. [24] . . . . . . . . . . . . . . . . 90 4.7 Braid, affine braid and virtual braid groups . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.7.1 Yang–Baxter groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.7.2 Some properties of braid and Yang–Baxter groups . . . . . . . . . . . . . . . . . . 97 4.7.3 Artin and Birman–Ko–Lee monoids . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5 Combinatorics of associative Yang–Baxter algebras 101 5.1 Combinatorics of Coxeter element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.1.1 Multiparameter deformation of Catalan, Narayana and Schröder numbers . . . . . 109 5.2 Grothendieck and q-Schröder polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.2.1 Schröder paths and polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.2.2 Grothendieck polynomials and k-dissections . . . . . . . . . . . . . . . . . . . . . . 114 5.2.3 Grothendieck polynomials and q-Schröder polynomials . . . . . . . . . . . . . . . . 115 5.2.4 Specialization of Schubert polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.2.5 Specialization of Grothendieck polynomials . . . . . . . . . . . . . . . . . . . . . . 133 5.3 The “longest element” and Chan–Robbins–Yuen polytope . . . . . . . . . . . . . . . . . . 134 5.3.1 The Chan–Robbins–Yuen polytope CRYn . . . . . . . . . . . . . . . . . . . . . . . 134 5.3.2 The Chan–Robbins–Mészáros polytope Pn,m . . . . . . . . . . . . . . . . . . . . . 139 5.4 Reduced polynomials of certain monomials . . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.4.1 Reduced polynomials, Motzkin and Riordan numbers . . . . . . . . . . . . . . . . 147 5.4.2 Reduced polynomials, dissections and Lagrange inversion formula . . . . . . . . . . 149 On Some Quadratic Algebras 3 A Appendixes 153 A.1 Grothendieck polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 A.2 Cohomology of partial flag varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 A.3 Multiparamater 3-term relations algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 A.3.1 Equivariant multiparameter 3-term relations algebras . . . . . . . . . . . . . . . . 159 A.3.2 Algebra 3QTn(β, h), generalized unitary case . . . . . . . . . . . . . . . . . . . . . 161 A.4 Koszul dual of quadratic algebras and Betti numbers . . . . . . . . . . . . . . . . . . . . . 162 A.5 On relations in the algebra Z0 n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 A.5.1 Hilbert series Hilb ( 3T 0 n , t ) and Hilb (( 3T 0 n )! , t ) : Examples . . . . . . . . . . . . . . 165 A.6 Summation and Duality transformation formulas [63] . . . . . . . . . . . . . . . . . . . . . 166 References 167 Extended abstract We introduce and study a certain class of quadratic algebras, which are nonhomogeneous in general, together with the distinguish set of mutually commuting elements inside of each, the so-called Dunkl elements. We describe relations among the Dunkl elements in the case of a family of quadratic algebras corresponding to a certain splitting of the universal classical Yang–Baxter relations into two three term relations. This result is a further extension and generalization of analogous results obtained in [45, 117] and [76]. As an application we describe explicitly the set of relations among the Gaudin elements in the group ring of the symmetric group, cf. [108]. We also study relations among the Dunkl elements in the case of (nonhomogeneous) quadratic algebras related with the universal dynamical classical Yang–Baxter relations. Some relations of results obtained in papers [45, 72, 75] with those obtained in [54] are pointed out. We also identify a subalgebra generated by the generators corresponding to the simple roots in the extended Fomin–Kirillov algebra with the DAHA, see Section 4.3. The set of generators of algebras in question, naturally corresponds to the set of edges of the complete graph Kn (to the set of edges and loops of the complete graph with (simple) loops K̃n in dynamical and equivariant cases). More generally, starting from any subgraph Γ of the complete graph with simple loops K̃n we define a (graded) subalgebra 3T (0) n (Γ) of the (graded) algebra 3T (0) n (K̃n) [70]. In the case of loop-less graphs Γ ⊂ Kn we state conjecture, Conjecture 4.15 in the main text, which relates the Hilbert polynomial of the abelian quotient 3T (0) n (Γ)ab of the algebra 3T (0) n (Γ) and the chromatic polynomial of the graph Γ we are started with12. We check 1We expect that a similar conjecture is true for any finite (oriented) matroid M. Namely, one (A.K.) can define an analogue of the three term relations algebra 3T (0)(M) for any (oriented) matroid M. We expect that the abelian quotient 3T (0)(M)ab of the algebra 3T (0)(M) is isomorphic to the Orlik–Terao algebra [114], denoted by OT(M) (known also as even version of the Orlik–Solomon algebra, denoted by OS+(M) ) associated with matroid M [28]. Moreover, the anticommutative quotient of the odd version of the algebra 3T (0)(M), as we expect, is isomorphic to the Orlik–Solomon algebra OS(M) associated with matroid M, see, e.g., [11, 49]. In particular, Hilb(3T (0)(M)ab, t ) = tr(M)Tutte ( M; 1 + t−1, 0 ) . We expect that the Tutte polynomial of a matroid, Tutte(M, x, y), is related with the Betti polynomial of a matroid M. Replacing relations u2 ij = 0, ∀ i, j, in the definition of the algebra 3T (0)(Γ) by relations u2 ij = qij , ∀ i, j, (i, j) ∈ E(Γ), where {qij}(i,j)∈E(Γ), qij = qji, is a collection of central elements, give rise to a quantization of the Orlik–Terao algebra OT(Γ). It seems an interesting task to clarify combinatorial/geometric significance of noncommutative versions of Orlik–Terao algebras (as well as Orlik–Solomon ones) defined as follows: OT (Γ) := 3T (0)(Γ), its “quantization” 3T (q)(Γ)ab and K-theoretic analogue 3T (q)(Γ, β)ab, cf. Definition 3.1, in the theory of hyperplane arrangements. Note that a small modification of arguments in [89] as were used for the proof of our Conjecture 4.15, gives rise to a theorem that the algebra 3Tn(Γ)ab is isomorphic to the Orlik–Terao algebra OT(Γ) studied in [126]. 2In the case of simple graphs our Conjecture 4.15 has been proved in [89]. 4 A.N. Kirillov our conjecture for the complete graphs Kn and the complete bipartite graphs Kn,m. Besides, in the case of complete multipartite graph Kn1,...,nr , we identify the commutative subalgebra in the algebra 3T (0) N (Kn1,...,nr), N = n1 + · · ·+ nr, generated by the elements θ (N) j,kj := ekj ( θ (N) Nj−1+1, . . . , θ (N) Nj ) , 1 ≤ j ≤ r, 1 ≤ kj ≤ nj , Nj := n1 + · · ·+ nj , N0 = 0, with the cohomology ring H∗(F ln1,...,nr ,Z) of the partial flag variety F ln1,...,nr . In other words, the set of (additive) Dunkl elements { θ (N) Nj−1+1, . . . , θ (N) Nj } plays a role of the Chern roots of the tau- tological vector bundles ξj , j = 1, . . . , r, over the partial flag variety F ln1,...,nr , see Section 4.1.2 for details. In a similar fashion, the set of multiplicative Dunkl elements { Θ (N) Nj−1+1, . . . ,Θ (N) Nj } plays a role of the K-theoretic version of Chern roots of the tautological vector bundle ξj over the partial flag variety F ln1,...,nr . As a byproduct for a given set of weights ` = {`ij}1≤i<j≤r we compute the Tutte polynomial T (K (`) n1,...,nk , x, y) of the `-weighted complete multipartite graph K (`) n1,...,nk , see Section 4, Definition 4.4 and Theorem 4.3. More generally, we introduce universal Tutte polynomial Tn({qij}, x, y) ∈ Z[{qij}][x, y] in such a way that for any collection of non-negative integers m = {mij}1≤i<j≤n and a subgraph Γ ⊂ K (m) n of the weighted complete graph on n labeled vertices with each edge (i, j) ∈ K(m) n appears with multiplicity mij , the specialization qij −→ 0 if edge (i, j) /∈ Γ, qij −→ [mij ]y := ymij − 1 y − 1 if edge (i, j) ∈ Γ of the universal Tutte polynomial is equal to the Tutte polynomial of graph Γ multiplied by (x− 1)κ(Γ), see Section 4.1.2, Theorem 4.24, and comments and examples, for details. We also introduce and study a family of (super) 6-term relations algebras, and suggest a definition of “multiparameter quantum deformation” of the algebra of the curvature of 2-forms of the Hermitian linear bundles over the complete flag variety F ln. This algebra can be treated as a natural generalization of the (multiparameter) quantum cohomology ring QH∗(F ln), see Section 4.2. In a similar fashion as in the case of three term relations algebras, for any sub- graph Γ ⊂ Kn, one (A.K.) can also define an algebra 6T (0)(Γ) and projection3 Ch: 6T (0)(Γ) −→ 3T (0)(Γ). Note that subalgebra A(Γ) := Q[θ1, . . . , θn] ⊂ 6T (0)(Γ)ab generated by additive Dunkl elements θi = ∑ j (ij)∈E(Γ) uij is closely related with problems have been studied in [118, 129], . . . , and [137] in the case Γ = Kn, see Section 4.2.2. We want to draw attention of the reader to the following problems related with arithmetic Schubert4 and Grothendieck calculi: (i) Describe (natural) quotient 6T †(Γ) of the algebra 6T (0)(Γ) such that the natural epi- morphism pr: A(Γ) −→ A(Γ) turns out to be isomorphism, where we denote by A(Γ) a subalgebra of 6T †(Γ) generated over Q by additive Dunkl elements. 3We treat this map as an algebraic version of the homomorphism which sends the curvature of a Hermitian vector bundle over a smooth algebraic variety to its cohomology class, as well as a splitting of classical Yang–Baxter relations (that is six term relations) in a couple of three term relations. 4See for example [137] and the literature quoted therein. On Some Quadratic Algebras 5 (ii) It is not difficult to see [72] that multiplicative Dunkl elements {Θi}1≤i≤n also mutu- ally commute in the algebra 6T (0), cf. Section 3.2. Problem we are interested in is to describe commutative subalgebras generated by multiplicative Dunkl elements in the alge- bras 6T †(Γ) and 6T (0)(Γ)ab. In the latter case one will come to the K-theoretic version of algebras studied in [118], . . . . Yet another objective of our paper5 is to describe several combinatorial properties of some special elements in the associative quasi-classical Yang–Baxter algebras [72], including among others, the so-called Coxeter element and the longest element. In the case of Coxeter element we relate the corresponding reduced polynomials introduced in [133, Exercise 6.C5(c)], and independently in [72], cf. [70], with the β-Grothendieck polynomials [42] for some special per- mutations π (n) k . More generally, we identify the β-Grothendieck polynomial G (β) π (n) k (Xn) with a certain weighted sum running over the set of k-dissections of a convex (n + k + 1)-gon. In particular we show that the specialization G (β) π (n) k (1) of the β-Grothendieck polynomial G (β) π (n) k (Xn) counts the number of k-dissections of a convex (n+k+1)-gon according to the number of diago- nals involved. When the number of diagonals in a k-dissection is the maximal possible (equals to n(2k−1)−1), we recover the well-known fact that the number of k-triangulations of a convex (n+k+1)-gon is equal to the value of a certain Catalan–Hankel determinant, see, e.g., [129]. In Section 5.4.2 we study multiparameter generalizations of reduced polynomials associated with Coxeter elements. We also show that for a certain 5-parameters family of vexillary permutations, the speciali- zation xi = 1, ∀ i ≥ 1, of the corresponding β-Schubert polynomials S (β) w (Xn) turns out to be coincide either with the Fuss–Narayana polynomials and their generalizations, or with a (q, β)- deformation of VSASM or that of CSTCPP numbers, see Corollary 5.33B. As examples we show that (a) the reduced polynomial corresponding to a monomial xn12x m 23 counts the number of (n,m)- Delannoy paths according to the number of NE-steps, see Lemma 5.81; (b) if β = 0, the reduced polynomial corresponding to monomial (x12x23)nxk34, n ≥ k, counts the number of n up, n down permutations in the symmetric group S2n+k+1, see Proposi- tion 5.82; see also Conjecture 5.83. We also point out on a conjectural connection between the sets of maximal compatible se- quences for the permutation σn,2n,2,0 and that σn,2n+1,2,0 from one side, and the set of VSASM(n) and that of CSTCPP(n) correspondingly, from the other, see Comments 5.48 for details. Finally, in Sections 5.1.1 and 5.4.1 we introduce and study a multiparameter generalization of reduced polynomials considered in [133, Exercise 6.C5(c)], as well as that of the Catalan, Narayana and (small) Schröder numbers. In the case of the longest element we relate the corresponding reduced polynomial with the Ehrhart polynomial of the Chan–Robbins–Yuen polytope, see Section 5.3. More generally, we relate the (t, β)-reduced polynomial corresponding to monomial n−1∏ j=1 x aj j,j+1 n−2∏ j=2  n∏ k=j+2 xjk  , aj ∈ Z≥0, ∀ j, 5This part of our paper had its origin in the study/computation of relations among the additive and multiplica- tive Dunkl elements in the quadratic algebras we are interested in, as well as the author’s attempts to construct a monomial basis in the algebra 3T (0) n and find its Hilbert series for n ≥ 6. As far as I’m aware these problems are still widely open. 6 A.N. Kirillov with positive t-deformations of the Kostant partition function and that of the Ehrhart polynomial of some flow polytopes, see Section 5.3. In Section 5.4 we investigate reduced polynomials associated with certain monomials in the algebra (ÂCYB)abn (β), known also as Gelfand–Varchenko algebra [67, 72], and study its combina- torial properties. Our main objective in Section 5.4.2 is to study reduced polynomials for Coxeter element treated in a certain multiparameter deformation of the (noncommutative) quadratic al- gebra ÂCYBn(α, β). Namely, to each dissection of a convex (n+ 2)-gon we associate a certain weight and consider the generating function of all dissections of (n+ 2)-gon selected taken with that weight. One can show that the reduced polynomial corresponding to the Coxeter element in the deformed algebra is equal to that generating function. We show that certain specializa- tions of that reduced polynomial coincide, among others, with the Grothendieck polynomials corresponding to the permutation 1 × w(n−1) 0 ∈ Sn, the Lagrange inversion formula, as well as give rise to combinatorial (i.e., positive expressions) multiparameters deformations of Catalan and Fuss–Catalan, Motzkin, Riordan and Fine numbers, Schröder numbers and Schröder trees. We expect (work in progress) a similar connections between Schubert and Grothendieck poly- nomials associated with the Richardson permutations 1k × w (n−k) 0 , k-dissections of a convex (n+ k + 1)-gon investigated in the present paper, and k-dimensional Lagrange–Good inversion formula studied from combinatorial point of view, e.g., in [22, 50]. 1 Introduction The Dunkl operators have been introduced in the later part of 80’s of the last century by Charles Dunkl [35, 36] as a powerful mean to study of harmonic and orthogonal polynomials related with finite Coxeter groups. In the present paper we don’t need the definition of Dunkl operators for arbitrary (finite) Coxeter groups, see, e.g., [35], but only for the special case of the symmetric group Sn. Definition 1.1. Let Pn = C[x1, . . . , xn] be the ring of polynomials in variables x1, . . . , xn. The type An−1 (additive) rational Dunkl operators D1, . . . , Dn are the differential-difference operators of the following form Di = λ ∂ ∂xi + ∑ j 6=i 1− sij xi − xj , (1.1) Here sij , 1 ≤ i < j ≤ n, denotes the exchange (or permutation) operator, namely, sij(f)(x1, . . . , xi, . . . , xj , . . . , xn) = f(x1, . . . , xj , . . . , xi, . . . , xn), ∂ ∂xi stands for the derivative w.r.t. the variable xi, λ ∈ C is a parameter. The key property of the Dunkl operators is the following result. Theorem 1.2 (C. Dunkl [35]). For any finite Coxeter group (W,S), where S = {s1, . . . , sl} denotes the set of simple reflections, the Dunkl operators Di := Dsi and Dj := Dsj pairwise commute: DiDj = DjDi, 1 ≤ i, j ≤ l. Another fundamental property of the Dunkl operators which finds a wide variety of applica- tions in the theory of integrable systems, see, e.g., [56], is the following statement: the operator l∑ i=1 (Di) 2 “essentially” coincides with the Hamiltonian of the rational Calogero–Moser model related to the finite Coxeter group (W,S). On Some Quadratic Algebras 7 Definition 1.3. Truncated (additive) Dunkl operator (or the Dunkl operator at critical level), denoted by Di, i = 1, . . . , l, is an operator of the form (1.1) with parameter λ = 0. For example, the type An−1 rational truncated Dunkl operator has the following form Di = ∑ j 6=i 1− sij xi − xj . Clearly the truncated Dunkl operators generate a commutative algebra. The important property of the truncated Dunkl operators is the following result discovered and proved by C. Dunkl [36]; see also [8] for a more recent proof. Theorem 1.4 (C. Dunkl [36], Yu. Bazlov [8]). For any finite Coxeter group (W,S) the algebra over Q generated by the truncated Dunkl operators D1, . . . ,Dl is canonically isomorphic to the coinvariant algebra AW of the Coxeter group (W,S). Recall that for a finite crystallographic Coxeter group (W,S) the coinvariant algebra AW is isomorphic to the cohomology ring H∗(G/B,Q) of the flag variety G/B, where G stands for the Lie group corresponding to the crystallographic Coxeter group (W,S) we started with. Example 1.5. In the case when W = Sn is the symmetric group, Theorem 1.4 states that the algebra over Q generated by the truncated Dunkl operators Di = ∑ j 6=i 1−sij xi−xj , i = 1, . . . , n, is canonically isomorphic to the cohomology ring of the full flag variety F ln of type An−1 Q[D1, . . . ,Dn] ∼= Q[x1, . . . , xn]/Jn, (1.2) where Jn denotes the ideal generated by the elementary symmetric polynomials {ek(Xn), 1 ≤ k ≤ n}. Recall that the elementary symmetric polynomials ei(Xn), i = 1, . . . , n, are defined through the generating function 1 + n∑ i=1 ei(Xn)ti = n∏ i=1 (1 + txi), where we set Xn := (x1, . . . , xn). It is well-known that in the case W = Sn, the isomorphism (1.2) can be defined over the ring of integers Z. Theorem 1.4 by C. Dunkl has raised a number of natural questions: (A) What is the algebra generated by the truncated • trigonometric, • elliptic, • super, matrix, . . . , (a) additive Dunkl operators? (b) Ruijsenaars–Schneider–Macdonald operators? (c) Gaudin operators? (B) Describe commutative subalgebra generated by the Jucys–Murphy elements in • the group ring of the symmetric group; • the Hecke algebra; • the Brauer algebra, BMV algebra, . . . . 8 A.N. Kirillov (C) Does there exist an analogue of Theorem 1.4 for • classical and quantum equivariant cohomology and equivariant K-theory rings of the partial flag varieties? • chomology and K-theory rings of affine flag varieties? • diagonal coinvariant algebras of finite Coxeter groups? • complex reflection groups? The present paper is an extended introduction to a few items from Section 5 of [72]. The main purpose of my paper “On some quadratic algebras, II” is to give some partial answers on the above questions basically in the case of the symmetric group Sn. The purpose of the present paper is to draw attention to an interesting class of nonho- mogeneous quadratic algebras closely connected (still mysteriously!) with different branches of Mathematics such as classical and quantum Schubert and Grothendieck calculi, low-dimensional topology, classical, basic and elliptic hypergeometric functions, algebraic combinatorics and graph theory, integrable systems, etc. What we try to explain in [72] is that upon passing to a suitable representation of the quadratic algebra in question, the subjects mentioned above, are a manifestation of certain general prop- erties of that quadratic algebra. From this point of view, we treat the commutative subalgebra generated (over a universal Lazard ring Ln [88]) by the additive (resp. multiplicative) truncated Dunkl elements in the algebra 3Tn(β), see Definition 3.1, as universal cohomology (resp. universal K-theory) ring of the complete flag variety F ln. The classical or quantum cohomology (resp. the classical or quantum K-theory) rings of the flag variety F ln are certain quotients of that universal ring. For example, in [74] we have computed relations among the (truncated) Dunkl elements {θi, i = 1, . . . , n} in the elliptic representation of the algebra 3Tn(β = 0). We expect that the commutative subalgebra obtained is isomorphic to elliptic cohomology ring (not defined yet, but see [48, 52]) of the flag variety F ln. Another example from [72]. Consider the algebra 3Tn(β = 0). One can prove [72] the following identities in the algebra 3Tn(β = 0): (A) summation formula n−1∑ j=1  n−1∏ b=j+1 ub,b+1 u1,n ( j−1∏ b=1 ub,b+1 ) = n−1∏ a=1 ua,a+1; (B) duality transformation formula, let m ≤ n, then n−1∑ j=m  n−1∏ b=j+1 ub,b+1 [m−1∏ a=1 ua,a+n−1ua,a+n ] um,m+n−1 ( j−1∏ b=m ub,b+1 ) + m∑ j=2 m−1∏ a=j ua,a+n−1ua,a+n um,n+m−1 ( n−1∏ b=m ub,b+1 ) u1,n = m∑ j=1 [ m−j∏ a=1 ua,a+nua+1,a+n ]( n−1∏ b=m ub,b+1 )[ j−1∏ a=1 ua,a+n−1ua,a+n ] . One can check that upon passing to the elliptic representation of the algebra 3Tn(β = 0), see Section 3.1 or [74], for the definition of elliptic representation, the above identities (A) On Some Quadratic Algebras 9 and (B) finally end up correspondingly, to be the summation formula and the N = 1 case of the duality transformation formula for multiple elliptic hypergeometric series (of type An−1), see, e.g., [63] or Appendix A.6 for the explicit forms of the latter. After passing to the so-called Fay representation [72], the identities (A) and (B) become correspondingly to be the summation formula and duality transformation formula for the Riemann theta functions of genus g > 0 [72]. These formulas in the case g ≥ 2 seems to be new. Worthy to mention that the relation (A) above can be treated as a “non-commutative ana- logue” of the well-known recurrence relation among the Catalan numbers. The study of “de- scendent relations” in the quadratic algebras in question was originally motivated by the author attempts to construct a monomial basis in the algebra 3T (0) n , and compute Hilb(3T (0) n , t) for n ≥ 6. These problems are still widely open, but gives rise the author to discovery of several interesting connections with • classical and quantum Schubert and Grothendieck calculi, • combinatorics of reduced decomposition of some special elements in the symmetric group, • combinatorics of generalized Chan–Robbins–Yuen polytopes, • relations among the Dunkl and Gaudin elements, • computation of Tutte and chromatic polynomials of the weighted complete multipartite graphs, etc. A few words about the content of the present paper. Example 1.5 can be viewed as an illustration of the main problems we are treated in Sections 2 and 3 of the present paper, namely the following ones. • Let {uij , 1 ≤ i, j ≤ n} be a set of generators of a certain algebra over a commutative ring K. The first problem we are interested in is to describe “a natural set of relations” among the generators {uij}1≤i,j≤n which implies the pair-wise commutativity of dynamical Dunkl elements θi = θ (n) i =: n∑ j=1 uij , 1 ≤ i ≤ n. • Should this be the case then we are interested in to describe the algebra generated by “the integrals of motions”, i.e., to describe the quotient of the algebra of polynomials K[y1, . . . , yn] by the two-sided ideal Jn generated by non-zero polynomials F (y1, . . . , yn) such that F (θ1, . . . , θn) = 0 in the algebra over ring K generated by the elements {uij}1≤i,j≤n. • We are looking for a set of additional relations which imply that the elementary symmetric polynomials ek(Yn), 1 ≤ k ≤ n, belong to the set of integrals of motions. In other words, the value of elementary symmetric polynomials ek(y1, . . . , yn), 1 ≤ k ≤ n, on the Dunkl elements θ (n) 1 , . . . , θ (n) n do not depend on the variables {uij , 1 ≤ i 6= j ≤ n}. If so, one can defined deformation of elementary symmetric polynomials, and make use of it and the Jacobi–Trudi formula, to define deformed Schur functions, for example. We try to realize this program in Sections 2 and 3. In Section 2, see Definition 2.3, we introduce the so-called dynamical classical Yang–Baxter algebra as “a natural quadratic algebra” in which the Dunkl elements form a pair-wise commuting family. It is the study of the algebra generated by the (truncated) Dunkl elements that is the main objective of our investigation in [72] and the present paper. In Section 2.1 we describe few representations of the dynamical classical Yang–Baxter algebra DCYBn related with 10 A.N. Kirillov • quantum cohomology QH∗(F ln) of the complete flag variety F ln, cf. [41]; • quantum equivariant cohomology QH∗Tn×C∗(T ∗F ln) of the cotangent bundle T ∗F ln to the complete flag variety, cf. [54]; • Dunkl–Gaudin and Dunkl–Uglov representations, cf. [108, 138]. In Section 3, see Definition 3.1, we introduce the algebra 3HTn(β), which seems to be the most general (noncommutative) deformation of the (even) Orlik–Solomon algebra of type An−1, such that it’s still possible to describe relations among the Dunkl elements, see Theorem 3.8. As an application we describe explicitly a set of relations among the (additive) Gaudin/Dunkl ele- ments, cf. [108]. It should be stressed at this place that we treat the Gaudin elements/operators (either additive or multiplicative) as images of the universal Dunkl elements/operators (addi- tive or multiplicative) in the Gaudin representation of the algebra 3HTn(0). There are seve- ral other important representations of that algebra, for example, the Calogero–Moser, Bruhat, Buchstaber–Felder–Veselov (elliptic), Fay trisecant (τ -functions), adjoint, and so on, considered (among others) in [72]. Specific properties of a representation chosen6 (e.g., Gaudin representa- tion) imply some additional relations among the images of the universal Dunkl elements (e.g., Gaudin elements) should to be unveiled. We start Section 3 with definition of algebra 3Tn(β) and its “Hecke” 3HTn(β) and “elliptic” 3MTn(β) quotients. In particular we define an elliptic representation of the algebra 3Tn(0) [74], and show how the well-known elliptic solutions of the quantum Yang–Baxter equation due to A. Belavin and V. Drinfeld, see, e.g., [9], S. Shibukawa and K. Ueno [130], and G. Felder and V. Pasquier [40], can be plug in to our construction, see Section 3.1. At the end of Section 3.1.1 we point out on a mysterious (at least for the author) appearance of the Euler numbers and “traces” of the Brauer algebra in the equivariant Pieri rules hold for the algebra 3TMn(β, q, ψ) stated in Theorem 3.8. In Section 3.2 we introduce a multiplicative analogue of the Dunkl elements {Θj ∈ 3Tn(β), 1 ≤ j ≤ n} and describe the commutative subalgebra in the algebra 3Tn(β) generated by mul- tiplicative Dunkl elements [76]. The latter commutative subalgebra turns out to be isomorphic to the quantum equivariant K-theory of the complete flag variety F ln [76]. In Section 3.3 we describe relations among the truncated Dunkl–Gaudin elements. In this case the quantum parameters qij = p2 ij , where parameters {pij = (zi − zj)−1, 1 ≤ i < j ≤ n} satisfy the both Arnold and Plücker relations. This observation has made it possible to describe a set of additional rational relations among the Dunkl–Gaudin elements, cf. [108]. In Section 3.4 we introduce an equivariant version of multiplicative Dunkl elements, called shifted Dunkl elements in our paper, and describe (some) relations among the latter. This result is a generalization of that obtained in Section 3.1 and [76]. However we don’t know any geometric interpretation of the commutative subalgebra generated by shifted Dunkl elements. In Section 4.1 for any subgraph Γ ⊂ Kn of the complete graph Kn we introduce7 [70, 72], algebras 3Tn(Γ) and 3T (0) n (Γ) which can be seen as analogues of algebras 3Tn and 3T (0) n corre- spondingly8. 6For example, in the cases of either Calogero–Moser or Bruhat representations one has an additional constraint, namely, u2 ij = 0 for all i 6= j. In the case of Gaudin representation one has an additional constraint u2 ij = p2 ij , where the (quantum) parameters {pij = 1 xi−xj , i 6= j}, satisfy simultaneously the Arnold and Plücker relations, see Section 2, II. Therefore, the (small) quantum cohomology ring of the type An−1 full flag variety F ln and the Bethe subalgebra(s) (i.e., the subalgebra generated by Gaudin elements in the algebra 3HTn(0)) correspond to different specializations of “quantum parameters” {qij := u2 ij} of the universal cohomology ring (i.e., the subalgebra/ring in 3HTn(0) generated by (universal) Dunkl elements). For more details and examples, see Section 2.1 and [72]. 7Independently the algebra 3T (0) n (Γ) has been studied in [16], where the reader can find some examples and conjectures. 8To avoid confusions, it must be emphasized that the defining relations for algebras 3Tn(Γ) and 3Tn(Γ)(0) may have more then three terms. On Some Quadratic Algebras 11 We want to point out in the Introduction, cf. footnote 1, that an analog of the algebras 3Tn and 3T (β) n , 3HTn, etc. treated in the present paper, can be defined for any (oriented or not) matroidM. We denote these algebras as 3T (M) and 3T (β)(M). One can show (A.K.) that the abelianization of the algebra 3T (β)(M), denoted by 3T (β)(M) ab , is isomorphic to the Gelfand– Varchenko algebra corresponding to a matroidM, whereas the algebra 3T (β=0)(M) ab is isomor- phic to the (even) Orlik–Solomon algebra OS+(M) of a matroidM.9 We consider and treat the algebras 3T (M), 3HT (M), . . . , as equivariant noncommutative (or quantum) versions of the (even) Orlik–Solomon algebras associated with matroid (including hyperplane, graphic, . . . ar- rangements). However a meaning of a quantum deformation of the (even or odd) Orlik–Solomon algebra suggested in the present paper, is missing, even for the braid arrangement of type An. Generalizations of the Gelfand–Varchenko algebra has been suggested and studied in [67, 72] and in the present paper under the name quasi-associative Yang–Baxter algebra, see Section 5. In the present paper we basically study the abelian quotient of the algebra 3T (0) n (Γ), where graph Γ has no loops and multiple edges, since we expect some applications of our approach to the theory of chromatic polynomials of planar graphs, in particular to the complete multipartite graphs Kn1,...,nr and the grid graphs Gm,n.10 Our main results hold for the complete multipartite, cyclic and line graphs. In particular we compute their chromatic and Tutte polynomials, see Proposition 4.19 and Theorem 4.24. As a byproduct we compute the Tutte polynomial of the `- weighted complete multipartite graph K (`) n1,...,nr where ` = {`ij}1≤i<j≤r, is a collection of weights, i.e., a set of non-negative integers. More generally, for a set of variables {{qij}1≤i<j≤n, x, y} we define universal Tutte polynomial Tn({qij}, x, y) ∈ Z[qij ][x, y] such that for any collection of non-negative integers {mij}1≤i<j≤n and a subgraph Γ ⊂ K(m) n of the complete graph Kn with each edge (i, j) comes with multiplic- ity mij , the specialization qij −→ 0 if edge (i, j) /∈ Γ, qij −→ [mij ]y := ymij − 1 y − 1 if edge (i, j) ∈ Γ of the universal Tutte polynomial Tn({qij}, x, y) is equal to the Tutte polynomial of graph Γ multiplied by the factor (t− 1)κ(Γ): (x− 1)κ(Γ)Tutte(Γ, x, y) := Tn({qij}, x, y) ∣∣∣∣ qij=0 if (i,j)/∈Γ qij=[mij ] y if (i,j)∈Γ . Here and after κ(Γ) demotes the number of connected components of a graph Γ. In other words, one can treat the universal Tutte polynomial Tn({qij}, x, y) as a “reproducing kernel” for the Tutte polynomials of all (loop-less) graphs with the number of vertices not exceeded n. We also state Conjecture 4.15 that for any loopless graph Γ (possibly with multiple edges) the algebra 3T (0) \|Γ\| (Γ) ab is isomorphic to the even Orlik–Solomon algebra OS+(AΓ) of the graphic arrangement associated with graph Γ in question11. At the end we emphasize that the case of the complete graph Γ = Kn reproduces the results of the present paper and those of [72], i.e., the case of the full flag variety F ln. The case of the complete multipartite graph Γ = Kn1,...,nr reproduces the analogue of results stated in the present paper for the case of full flag variety F ln, to the case of the partial flag variety Fn1,...,nr , see [72] for details. 9For a definition and basic properties of the Orlik–Solomon algebra corresponding to a matroid, see, e.g., [49, 65]. 10See http://reference.wolfram.com/language/ref/GridGraph.html for a definition of grid graph Gm,n. 11For simple graphs, i.e., without loops and multiple edges, this conjecture has been proved in [89]. http://reference.wolfram.com/language/ref/GridGraph.html 12 A.N. Kirillov In Section 4.1.4 we sketch how to generalize our constructions and some of our results to the case of the Lie algebras of classical types12. In Section 4.2 we briefly overview our results concerning yet another interesting family of quadratic algebras, namely the six-term relations algebras 6Tn, 6T (0) n and related ones. These algebras also contain a distinguished set of mutually commuting elements called Dunkl elements {θi, i = 1, . . . , n} given by θi = ∑ j 6=i rij , see Definition 4.48. In Section 4.2.2 we introduce and study the algebra 6TFn in greater detail. In particular we introduce a “quantum deformation” of the algebra generated by the curvature of 2-forms of of the Hermitian linear bundles over the flag variety F ln, cf. [118]. In Section 4.2.3 we state our results concerning the classical Yang–Baxter algebra CYBn and the 6-term relation algebra 6Tn. In particular we give formulas for the Hilbert series of these algebras. These formulas have been obtained independently in [7] The paper just mentioned, contains a description of a basis in the algebra 6Tn, and much more. In Section 4.2.4 we introduce a super analog of the algebra 6Tn, denoted by 6Tn,m, and compute its Hilbert series. Finally, in Section 4.3 we introduce extended nil-three term relations algebra 3Tn and describe a subalgebra inside of it which is isomorphic to the double affine Hecke algebra of type An−1, cf. [24]. In Section 5 we describe several combinatorial properties of some special elements in the associative quasi-classical Yang–Baxter algebra13, denoted by ÂCYBn. The main results in that direction were motivated and obtained as a by-product, in the process of the study of the the structure of the algebra 3HTn(β). More specifically, the main results of Section 5 were obtained in the course of “hunting for descendant relations” in the algebra mentioned, which is an important problem to be solved to construct a basis in the nil-quotient algebra 3T (0) n . This problem is still widely-open. The results of Section 5.1, see Proposition 5.4, items (1)–(5), are more or less well-known among the specialists in the subject, while those of the item (6) seem to be new. Namely, we show that the polynomial Qn(xij = ti) from [133, Exercise 6.C8(c)], essentially coincides with the β-deformation [42] of the Lascoux–Schützenberger Grothendieck polynomial [86] for some particular permutation. The results of Proposition 5.4(6), point out on a deep connection between reduced forms of monomials in the algebra ÂCYBn and the Schubert and Grothendieck calculi. This observation was the starting point for the study of some combinatorial properties of certain specializations of the Schubert, the β-Grothendieck [43] and the double β-Grothendieck polynomials in Section 5.2. One of the main results of Section 5.2 can be stated as follows. Theorem 1.6. (1) Let w ∈ Sn be a permutation, consider the specialization x1 := q, xi = 1, ∀ i ≥ 2, of the β-Grothendieck polynomial G (β) w (Xn). Then Rw(q, β + 1) := G(β) w (x1 = q, xi = 1, ∀ i ≥ 2) ∈ N[q, 1 + β]. In other words, the polynomial Rw(q, β) has non-negative integer coefficients14. For late use we define polynomials Rw(q, β) := q1−w(1)Rw(q, β). 12One can define an analogue of the algebra 3T (0) n for the root system of BCn and C∨nCn-types as well, but we are omitted these cases in the present paper. 13The algebra ÂCYBn can be treated as “one-half” of the algebra 3Tn(β). It appears that the basic relations among the Dunkl elements, which do not mutually commute anymore, are still valid, see Lemma 5.3. 14For a more general result see Appendix A.1, Corollary A.7. On Some Quadratic Algebras 13 (2) Let w ∈ Sn be a permutation, consider the specialization xi := q, yi = t, ∀ i ≥ 1, of the double β-Grothendieck polynomial G (β) w (Xn, Yn). Then G(β−1) w (xi := q, yi := t, ∀ i ≥ 1) ∈ N[q, t, β]. (3) Let w be a permutation, then Rw(1, β) = R1×w(0, β). Note that Rw(1, β) = Rw−1(1, β), but Rw(t, β) 6= Rw−1(t, β), in general. For the reader convenience we collect some basic definitions and results concerning the β- Grothendieck polynomials in Appendix A.1. Let us observe that Rw(1, 1) = Sw(1), where Sw(1) denotes the specialization xi := 1, ∀ i ≥ 1, of the Schubert polynomial Sw(Xn) corresponding to permutation w. Therefore, Rw(1, 1) is equal to the number of compatible sequences [13] (or pipe dreams, see, e.g., [129]) corresponding to permutation w. Problem 1.7. Let w ∈ Sn be a permutation and l := `(w) be its length. Denote by CS(w) = {a = (a1 ≤ a2 ≤ · · · ≤ al) ∈ Nl} the set of compatible sequences [13] corresponding to permutation w. • Define statistics r(a) on the set of all compatible sequences CSn := ∐ w∈Sn CS(w) in a such way that∑ a∈CS(w) qa1βr(a) = Rw(q, β). • Find a geometric interpretation, and investigate combinatorial and algebra-geometric pro- perties of polynomials S (β) w (Xn), where for a permutation w ∈ Sn we denoted by S (β) w (Xn) the β-Schubert polynomial defined as follows S(β) w (Xn) = ∑ a∈CS(w) βr(a) l:=`(w)∏ i=1 xai . We expect that polynomial S (β) w (1) coincides with the Hilbert polynomial of a certain graded commutative ring naturally associated to permutation w. Remark 1.8. It should be mentioned that, in general, the principal specialization G(β−1) w ( xi := qi−1, ∀ i ≥ 1 ) of the (β − 1)-Grothendieck polynomial may have negative coefficients. Our main objective in Section 5.2 is to study the polynomials Rw(q, β) for a special class of permutations in the symmetric group S∞. Namely, in Section 5.2 we study some combinatorial properties of polynomials R$λ,φ(q, β) for the five parameters family of vexillary permutations {$λ,φ} which have the shape λ := λn,p,b = (p(n− i+1)+ b, i = 1, . . . , n+1) and flag φ := φk,r = (k + r(i− 1), i = 1, . . . , n+ 1). This class of permutations is notable for many reasons, including that the specialized value of the Schubert polynomial S$λ,φ(1) admits a nice product formula15, see Theorem 5.29. Moreover, 15One can prove a product formula for the principal specialization S$λ,φ(xi := qi−1, ∀ i ≥ 1) of the correspon- ding Schubert polynomial. We don’t need a such formula in the present paper. 14 A.N. Kirillov we describe also some interesting connections of polynomials R$λ,φ(q, β) with plane partitions, the Fuss–Catalan numbers16 and Fuss–Narayana polynomials, k-triangulations and k-dissections of a convex polygon, as well as a connection with two families of ASM. For example, let λ = (bn) and φ = (kn) be rectangular shape partitions, then the polynomial R$λ,φ(q, β) defines a (q, β)- deformation of the number of (ordinary) plane partitions17 sitting in the box b×k×n. It seems an interesting problem to find an algebra-geometric interpretation of polynomials Rw(q, β) in the general case. Question 1.9. Let a and b be mutually prime positive integers. Does there exist a family of permutations wa,b ∈ Sab(a+b) such that the specialization xi = 1, ∀ i of the Schubert polyno- mial Swa,b is equal to the rational Catalan number Ca/b? That is Swa,b(1) = 1 a+ b ( a+ b a ) . Many of the computations in Section 5.2 are based on the following determinantal formula for β-Grothendieck polynomials corresponding to grassmannian permutations, cf. [84]. Theorem 1.10 (see Comments 5.37(b)). If w = σλ is the grassmannian permutation with shape λ = (λ, . . . , λn) and a unique descent at position n, then18 (A) G(β) σλ (Xn) = DET ∣∣h(β) λj+i,j (Xn) ∣∣ 1≤i,j≤n = DET ∣∣xλj+n−ji (1 + βxi) j−1 ∣∣ 1≤i,j≤n∏ 1≤i<j≤n (xi − xj) , where Xn = (xi, x1, . . . , xn), and for any set of variables X, h (β) n,k(X) = k−1∑ a=0 ( k − 1 a ) hn−k+a(X)βa, and hk(X) denotes the complete symmetric polynomial of degree k in the variables from the set X. (B) Gσλ(X,Y ) = DET ∣∣∣ λj+n−j∏ a=1 (xi + ya + βxiya)(1 + βxi) j−1 ∣∣∣ 1≤i,j≤n∏ 1≤i<j≤n (xi − xj) . 16We define the (generalized) Fuss–Catalan numbers to be FC (p) n (b) := 1+b 1+b+(n−1)p ( np+b n ) . Connection of the Fuss–Catalan numbers with the p-ballot numbers Balp(m,n) := n−mp+1 n+m+1 ( n+m+1 m ) and the Rothe numbers Rn(a, b) := a a+bn ( a+bn n ) can be described as follows FC(p) n (b) = Rn(b+ 1, p) = Balp−1(n, (n− 1)p+ b). 17Let λ be a partition. An ordinary plane partition (plane partition for short)bounded by d and shape λ is a filling of the shape λ by the numbers from the set {0, 1, . . . , d} in such a way that the numbers along columns and rows are weakly decreasing. A reverse plane partition bounded by d and shape λ is a filling of the shape λ by the numbers from the set {0, 1, . . . , d} in such a way that the numbers along columns and rows are weakly increasing. 18The equality G(β) σλ (Xn) = DET ∣∣xλj+n−ji (1 + βxi) j−1 ∣∣ 1≤i,j≤n∏ 1≤i<j≤n (xi − xj) , has been proved independently in [107]. On Some Quadratic Algebras 15 In Sections 5.2.2 and 5.4.2 we study connections of Grothendieck polynomial associated with the Richardson permutation w (n) k = 1k × w (n−k) 0 , k-dissections of a convex (n + k + 1)-gon, generalized reduced polynomial corresponding to a certain monomial in the algebra ÂCYBn and the Lagrange inversion formula. In the case of generalized Richardson permutation w (k) n,p corresponding to the k-shifted dominant permutations w(p,n) associated with the Young diagram λp,n := p(n−1, n−2, . . . , 1), namely, w (k) n,p = 1k×w(p,n), we treat only the case k = 1, see also [39]. In the case k ≥ 2 one comes to a task to count and find a lattice path type interpretation for the number of k-pgulations of a convex n-gon that is the number of partitioning of a convex n-gon on parts which are all equal to a convex (p+ 2)-gon, by a (maximal) family of diagonals such that each diagonal has at most k internal intersections with the members of a family of diagonals selected. In Section 5.3 we give a partial answer on Question 6.C8(d) by R. Stanley [133]. In particular, we relate the reduced polynomial corresponding to monomial ( xa2 12 · · ·xn−1,n an ) n−2∏ j=2 n∏ k=j+2 xjk, aj ∈ Z≥0, ∀ j, with the Ehrhart polynomial of the generalized Chan–Robbins–Yuen polytope, if a2 = · · · = an = m+ 1, cf. [101], with a t-deformation of the Kostant partition function of type An−1 and the Ehrhart polynomials of some flow polytopes, cf. [103]. In Section 5.4 we investigate certain specializations of the reduced polynomials corresponding to monomials of the form xm1 12 · · ·x mn n−1,n, mj ∈ Z≥0, ∀ j. First of all we observe that the corresponding specialized reduced polynomial appears to be a piece-wise polynomial function of parameters m = (m1, . . . ,mn) ∈ (R≥0)n, denoted by Pm. It is an interesting problem to compute the Laplace transform of that piece-wise polynomial function. In the present paper we compute the value of the function Pm in the dominant chamber Cn = (m1 ≥ m2 ≥ · · · ≥ mn ≥ 0), and give a combinatorial interpretation of the values of that function in points (n,m) and (n,m, k), n ≥ m ≥ k. For the reader convenience, in Appendices A.1–A.6 we collect some useful auxiliary informa- tion about the subjects we are treated in the present paper. Almost all results in Section 5 state that some two specific sets have the same number of elements. Our proofs of these results are pure algebraic. It is an interesting problem to find bijective proofs of results from Section 5 which generalize and extend remarkable bijective proofs presented in [103, 129, 135, 142] to the cases of • the β-Grothendieck polynomials, • the (small) Schröder numbers, • k-dissections of a convex (n+ k + 1)-gon, • special values of reduced polynomials. We are planning to treat and present these bijections in separate publication(s). We expect that the reduced polynomials corresponding to the higher-order powers of the Coxeter elements also admit an interesting combinatorial interpretation(s). Some preliminary results in this direction are discussed in Comments 5.67. At the end of introduction I want to add a few remarks. (a) After a suitable modification of the algebra 3HTn, see [75], and the case β 6= 0 in [72], one can compute the set of relations among the (additive) Dunkl elements (defined in Section 2, 16 A.N. Kirillov equation (2.1)). In the case β = 0 and qij = qiδj−i,1, 1 ≤ i < j ≤ n, where δa,b is the Kronecker delta symbol, the commutative algebra generated by additive Dunkl elements (2.3) appears to be “almost” isomorphic to the equivariant quantum cohomology ring of the flag variety F ln, see [75] for details. Using the multiplicative version of Dunkl elements, see Section 3.2, one can extend the results from [75] to the case of equivariant quantum K-theory of the flag variety F ln, see [72]. (b) As it was pointed out previously, one can define an analogue of the algebra 3T (0) n for any (oriented) matroid Mn, and state a conjecture which connects the Hilbert polynomial of the algebra 3T (0) n ((Mn)ab, t) and the chromatic polynomial of matroidMn. We expect that algebra 3T (β=1) n (Mn)ab is isomorphic to the Gelfand–Varchenko algebra associated with matroid M. It is an interesting problem to find a combinatorial meaning of the algebra 3T (β) n (Mn) for β = 0 and β 6= 0. (c) Let R be a (graded) ring (to be specified later) and Fn2 be the free associative algebra over R with the set of generators {uij , 1 ≤ i, j ≤ n}. In the subsequent text we will distinguish the set of generators {uii}1≤i≤n from that {uij}1≤i 6=j≤n, and set xi := uii, i = 1, . . . , n. A guiding idea to choose definitions and perform constructions in the present paper is to impose a set of relations Rn among the generators {xi}1≤i≤n and that {uij}1≤i 6=j≤n which ensure the mutual commutativity of the following elements θ (n) i := θi = xi + n∑ j 6=i uij , i = 1, . . . , n, in the algebra Fn2/Rn, as well as to have a good chance to describe/compute • “Integral of motions”, that is finding a big enough set of algebraically independent polyno- mials (quite possibly that polynomials are trigonometric or elliptic ones) I (n) α (y1, . . . , yn) ∈ R[Yn] such that I(n) α ( θ (n) 1 , . . . , θ(n) n ) ∈ R[Xn], ∀α, in other words, the latter specialization of any integral of motion has to be independent of the all generators {uij}1≤i 6=j≤n. • Give a presentation of the algebra In generated by the integral of motions that is to find a set of defining relations among the elements θ1, . . . , θn, and describe a R-basis { m (n) α } in the algebra In. • Generalized Littlewood–Richardson and Murnaghan–Nakayama problems. Given an inte- gral of motion I (m) β (Ym) and an integer n ≥ m, find an explicit positive (if possible) expression in the quotient algebra Fn2/Rn of the element I (m) β ( θ (n) 1 , . . . , θ(n) m ) . For example in the case of the 3-term relations algebra 3T (0) n (as well as its equivariant, quantum, etc. versions) the generalized Littlewood–Richardson problem is to find a positive expression in the algebra 3T (0) n for the element Sw ( θ (n) 1 , . . . , θ (n) m ) , where Sw(Yn) stands for the Schubert polynomial corresponding to a permutation w ∈ Sn. Generalized Murnaghan–Nakayama problem consists in finding a combinatorial expression in the algebra 3T (0) n for the element m∑ i=1 (θ (n) i )k. On Some Quadratic Algebras 17 Partial results concerning these problems have been obtained as far as we aware in [45, 70, 72, 73, 104, 117]. • “Partition functions”. Assume that the (graded) algebra In generated over R by the elements θ1, . . . , θn has finite dimension/rank, and the (non zero) maximal degree component I(n) max of that algebra has dimension/rank one and generated by an element ω. For any element g ∈ Fn2 let us denote by Resω(g) an element in R such that g = Resω(g)ω, where we denote by g the image of element g in the component I(n) max. We define partition function associated with the algebra In as follows Z(In) = Resw ( exp (∑ α qαm (n) α )) , where {qα} is a set of parameters which is consistent in one-to-one correspondence with a basis{ m (n) α } chosen. We are interesting in to find a closed formula for the partition function Z(In) as well as that for a small partition function Z(0)(In) := Resω ( exp ( ∑ 1≤i,j≤n λijuij )) , where {λij}1≤i,j≤n stands for a set of parameters. One can show [68] that the partition func- tion Z(In) associated with algebra 3T qn satisfies the famous Witten–Dijkraaf–Verlinde–Verlinde equations. As a preliminary steps to perform our guiding idea we (i) investigate properties of the abelianization of the algebra Fn2/Rn. Some unexpected connections with the theory of hyperplane arrangements and graph theory are discovered; (ii) investigate a variety of descendent relations coming from the defining relations. Some polynomials with interesting combinatorial properties are naturally appear. To keep the size of the present paper reasonable, several new results are presented as exercises. We conclude Introduction by a short historical remark. As far as we aware, the commu- tative version of 3-term relations which provided the framework for a definition of the FK algebra En [45] and a plethora of its generalizations, have been frequently used implicitly in the theory of elliptic functions and related topics, starting at least from the middle of the 19th century, see, e.g., [141] for references, and up to now, and for sure will be used for ever. The key point is that the Kronecker sigma function σz(w) := σ(z − w)θ′(0) σ(z)σ(−w) , where σ(z) denotes the Weierstrass sigma function, satisfies the quadratic three terms addition formula or functional equation discovered, as far as we aware, by K. Weierstrass. In fact this functional equation is really equivalent19 to the famous Jacobi–Riemann three term relation of degree four between the Riemann theta functions θ(x). In the rational degeneration of theta functions, the three term relation between Kronecker sigma functions turns to the famous three term Jacobi identity which can be treated as an associative analogue of the Jacobi identity in the theory of Lie algebras. 19We refer the reader to a nice written paper by Tom H. Koornwinder [79] for more historical information. 18 A.N. Kirillov To our best knowledge, in an abstract form that is as a set of defining relations in a certain algebra, an anticommutative version of three term relations had been appeared in a remarkable paper by V.I. Arnold [3]. Nowadays these relations are known as Arnold relations. These relations and its various generalizations play fundamental role in the theory of arrangements, see, e.g., [113], in topology, combinatorics and many other branches of Mathematics. In commutative set up abstract form of 3-term relations has been invented by O. Mathieu [96]. In the context of the braided Hopf algebras (of type A) 3-term relations like algebras (as some examples of the Nichols algebras) have appeared in papers by A. Milinski and H.-J. Schneider (2000), N. Andruskiewitsch (2002), S. Madjid (2004), I. Heckenberger (2005) and many others20. It is well-known that the Nichols algebra associated with the symmetric group Sn and trivial conjugacy class is a quotient of the algebra FKn. It is still an open problem to prove (or disprove) that these two algebras are isomorphic. 2 Dunkl elements Having in mind to fulfill conditions suggested by our guiding line mentioned in Introduction as far as it could be done till now, we are led to introduce the following algebras21. Definition 2.1 (additive Dunkl elements). The (additive) Dunkl elements θi, i = 1, . . . , n, in the algebra Fn are defined to be θi = xi + n∑ j=1 j 6=i uij . (2.1) We are interested in to find “natural relations” among the generators {uij}1≤i,j≤n such that the Dunkl elements (2.1) are pair-wise commute. One of the natural conditions which is the commonly accepted in the theory of integrable systems, is • locality conditions: (a) [xi, xj ] = 0 if i 6= j, (b) uijukl = ukluij if i 6= j, k 6= l and {i, j} ∩ {k, l} = ∅. (2.2) Lemma 2.2. Assume that elements {uij} satisfy the locality condition (2.1). If i 6= j, then [θi, θj ] = [ xi + ∑ k 6=i,j uik, uij + uji ] + [ uij , n∑ k=1 xk ] + ∑ k 6=i,j wijk, where wijk = [uij , uik + ujk] + [uik, ujk] + [xi, ujk] + [uik, xj ] + [xk, uij ]. (2.3) Therefore in order to ensure that the Dunkl elements form a pair-wise commuting family, it’s natural to assume that the following conditions hold 20We refer the reader to the site https://en.wikipedia.org/wiki/Nichols_algebra for basic definitions and results concerning Nichols’ algebras and references on vast literature treated different aspects of the theory of Nichols’ algebras and braided Hopf algebras. 21Surprisingly enough, in many cases to find relations among the elements θ1, . . . , θn there is no need to require that the elements {θi}1≤i≤n are pairwise commute. https://en.wikipedia.org/wiki/Nichols_algebra On Some Quadratic Algebras 19 • unitarity: [uij + uji, ukl] = 0 = [uij + uji, xk] for all distinct i, j, k, l, (2.4) i.e., the elements uij + uji are central. • “conservation laws”:[ n∑ k=1 xk, uij ] = 0 for all i, j, (2.5) i.e., the element E := n∑ k=1 xk is central, • unitary dynamical classical Yang–Baxter relations: [uij , uik + ujk] + [uik, ujk] + [xi, ujk] + [uik, xj ] + [xk, uij ] = 0, (2.6) if i, j, k are pair-wise distinct. Definition 2.3 (dynamical six term relations algebra 6DTn). We denote by 6DTn (and fre- quently will use also notation DCYBn) the quotient of the algebra Fn by the two-sided ideal generated by relations (2.2)–(2.6). Clearly, the Dunkl elements (2.1) generate a commutative subalgebra inside of the alge- bra 6DTn, and the sum n∑ i=1 θi = n∑ i=1 xi belongs to the center of the algebra 6DTn. Remark 2.4. Occasionally we will call the Dunkl elements of the form (2.1) by dynamical Dunkl elements to distinguish the latter from truncated Dunkl elements, corresponding to the case xi = 0, ∀ i. 2.1 Some representations of the algebra 6DTn 2.1.1 Dynamical Dunkl elements and equivariant quantum cohomology (I) ( cf. [41]). Given a set q1, . . . , qn−1 of mutually commuting parameters, define qij = j−1∏ a=i qa if i < j, and set qij = qji in the case i > j. Clearly, that if i < j < k, then qijqjk = qik. Let z1, . . . , zn be a set of (mutually commuting) variables. Denote by Pn := Z[z1, . . . , zn] the corresponding ring of polynomials. We consider the variable zi, i = 1, . . . , n, also as the operator acting on the ring of polynomials Pn by multiplication on the variable zi. Let sij ∈ Sn be the transposition that swaps the letters i and j and fixes the all other letters k 6= i, j. We consider the transposition sij also as the operator which acts on the ring Pn by interchanging zi and zj , and fixes all other variables. We denote by ∂ij = 1− sij zi − zj , ∂i := ∂i,i+1, the divided difference operators corresponding to the transposition sij and the simple transpo- sition si := si,i+1 correspondingly. Finally we define operator (cf. [41]) ∂(ij) := ∂i · · · ∂j−1∂j∂j−1 · · · ∂i if i < j. The operators ∂(ij), 1 ≤ i < j ≤ n, satisfy (among other things) the following set of relations (cf. [41]) 20 A.N. Kirillov • [zj , ∂(ik)] = 0 if j /∈ [i, k], [ ∂(ij), j∑ a=i za ] = 0, • [∂(ij), ∂(kl)] = δjk[zj , ∂(il)] + δil[∂(kj), zi] if i < j, k < l. Therefore, if we set uij = qij∂(ij) if i < j, and uij = −uji if i > j, then for a triple i < j < k we will have [uij , uik + ujk] + [uik, ujk] + [zi, ujk] + [uik, zj ] + [zk, ujk] = qijqjk[∂(ij), ∂(jk)] + qik[∂(ik), zj ] = 0. Thus the elements {zi, i = 1, . . . , n} and {uij , 1 ≤ i < j ≤ n} define a representation of the algebra DCYBn, and therefore the Dunkl elements θi := zi + ∑ j 6=i uij = zi − ∑ j<i qji∂(ji) + ∑ j>i qij∂(ij) form a pairwise commuting family of operators acting on the ring of polynomials Z[q1, . . . , qn−1][z1, . . . , zn], cf. [41]. This representation has been used in [41] to construct the small quantum cohomology ring of the complete flag variety of type An−1. (II) Consider degenerate affine Hecke algebra Hn generated by the central element h, the elements of the symmetric group Sn, and the mutually commuting elements y1, . . . , yn, subject to relations siyi − yi+1si = h, 1 ≤ i < n, siyj = yjsi, j 6= i, i+ 1, where si stand for the simple transposition that swaps only indices i and i + 1. For i < j, let sij = si · · · sj−1sjsj−1 · · · si denotes the permutation that swaps only indices i and j. It is an easy exercise to show that • [yj , sik] = h[sij , sjk] if i < j < k, • yisik − sikyk = h+ hsik ∑ i<j<k sjk if i < k. Finally, consider a set of mutually commuting parameters {pij , 1 ≤ i 6= j ≤ n, pij + pji = 0}, subject to the constraints pijpjk = pikpij + pjkpik + hpik, i < j < k. Comments 2.5. If parameters {pij} are invertible, and satisfy relations pijpjk = pikpij + pjkpik + βpik, i < j < k, then one can rewrite the above displayed relations in the following form 1 + β pik = ( 1 + β pij )( 1 + β pjk ) , 1 ≤ i < j < k ≤ n. Therefore there exist parameters {q1, . . . , qn} such that 1+β/pij = qi/qj , 1 ≤ i < j ≤ n. In other words, pij = βqj qj−qj , 1 ≤ i < j ≤ n. However in general, there are many other types of solutions, for example, solutions related to the Heaviside function22 H(x), namely, pij = H(xi − xj), xi ∈ R, ∀ i, and its discrete analogue, see Example (III) below. In the both cases β = −1; see also Comments 2.12 for other examples. 22See https://en.wikipedia.org/wiki/Heaviside_step_function. https://en.wikipedia.org/wiki/Heaviside_step_function On Some Quadratic Algebras 21 To continue presentation of Example (II), define elements uij = pijsij , 1 ≤ i 6= j ≤ n. Lemma 2.6 (dynamical classical Yang–Baxter relations). [uij , uik + ujk] + [uik, ujk] + [uik, yj ] = 0, 1 < i < j < k ≤ n. (2.7) Indeed, uijujk = uikuij + ujkuik + hpiksijsjk, ujkuij = uijuik + uikujk + hpiksjksij , and moreover, [yj , uik] = hpik[sij , sjk]. Therefore, the elements θi = yi − h ∑ j<i uij + h ∑ i<j uij , i = 1, . . . , n, form a mutually commuting set of elements in the algebra Z[{pij}]⊗Z Hn. Theorem 2.7. Define matrix Mn = (mi,j)1≤i,j≤n as follows mi,j(u; z1, . . . , zn) =  u− zi if i = j, −h− pij if i < j, pij if i > j. Then DET ∣∣Mn(u; θ1, . . . , θn) ∣∣ = n∏ j=1 (u− yj). Moreover, let us set qij := h2(pij + p2 ij) = h2qiqj(qi − qj)−2, i < j, then ek(θ1, . . . , θn) = e (q) k (y1, . . . , yn), 1 ≤ k ≤ n, where ek(x1, . . . , xn) and e (q) k (x1, . . . , xn) denote correspondingly the classical and multiparameter quantum [45] elementary polynomials23. Let’s stress that the elements yi and θj do not commute in the algebra Hn, but the symmetric functions of y1, . . . , yn, i.e., the center of the algebra Hn, do. A few remarks in order. First of all, u2 ij = p2 ij are central elements. Secondly, in the case h = 0 and yi = 0, ∀ i, the equality DET ∣∣Mn(u;x1, . . . , xn) ∣∣ = un describes the set of polynomial relations among the Dunkl–Gaudin elements (with the following choice of parameters pij = (qi− qj)−1 are taken). And our final remark is that according to [54, Section 8], the quotient ring Hqn := Q[y1, . . . , yn]Sn ⊗Q[θ1, . . . , θn]⊗Q[h] /〈 Mn(u; θ1, . . . , θn) = n∏ j=1 (u− yj) 〉 23For the reader convenience we remind [45] a definition of the quantum elementary polynomial eqk(x1, . . . , xn). Let q := {qij}1≤i<j≤n be a collection of “quantum parameters”, then eqk(x1, . . . , xn) = ∑ ` ∑ 1≤i1<···<i`≤n j1>i1,...,j`>i` ek−2`(XI∪J) ∏̀ a=1 qia,ja , where I = (i1, . . . , i`), J = (j1, . . . , j`) should be distinct elements of the set {1, . . . , n}, and XI∪J denotes set of variables xa for which the subscript a is neither one of im nor one of the jm. 22 A.N. Kirillov is isomorphic to the quantum equivariant cohomology ring of the cotangent bundle T ∗F ln of the complete flag variety of type An−1, namely, Hqn ∼= QH∗Tn×C∗(T ∗F ln) with the following choice of quantum parameters: Qi := hqi+1/qi, i = 1, . . . , n− 1. On the other hand, in [75] we computed the so-called multiparameter deformation of the equivariant cohomology ring of the complete flag variety of type An−1. A deformation defined in [75] depends on parameters {qij , 1 ≤ i < j ≤ n} without any constraints are imposed. For the special choice of parameters qij := h2 qi qj (qi − qj)2 the multiparameter deformation of the equivariant cohomology ring of the type An−1 com- plete flag variety F ln constructed in [75], is isomorphic to the ring Hqn. Comments 2.8. Let us fix a set of independent parameters {q1, . . . , qn} and define new pa- rameters{ qij := hpij(pij + h) = h2 qiqj (qi − qj)2 } , 1 ≤ i < j ≤ n, where pij = qj qi − qj , i < j. We set deg(qij) = 2, deg(pij) = 1, deg(h) = 1. The new parameters {qij}1≤i<j≤n, do not free anymore, but satisfy rather complicated alge- braic relations. We display some of these relations soon, having in mind a question: is there some intrinsic meaning of the algebraic variety defined by the set of defining relations among the “quantum parameters” {qij}? Let us denote by An,h the quotient ring of the ring of polynomials Q[h][xij , 1 ≤ i < j ≤ n] modulo the ideal generating by polynomials f(xij) such that the specialization xij = qij of a polynomial f(xij), namely f(qij), is equal to zero. The algebra An,h has a natural filtration, and we denote by An = grAn,h the corresponding associated graded algebra. To describe (a part of) relations among the parameters {qij} let us observe that parame- ters {pij} and {qij} are related by the following identity qijqjk − qik(qij + qjk) + h2qik = 2pijpikpjk(pik + h) if i < j < k. Using this identity we can find the following relations among parameters in question q2 ijq 2 jk + q2 ijq 2 ik + h4q2 ikq 2 jk − 2qijqikqjk(qij + qjk + qik) − 2h2qik(qijqjk + qijqik + qjkqik) = 8hqijqikqjkpik, (2.8) if 1 ≤ i < j < k ≤ n. Finally, we come to a relation of degree 8 among the “quantum parameters” {qij}( l.h.s. of (2.8) )2 = 64h2q2 ijq 3 ikq 2 jk, 1 ≤ i < j < k ≤ n. There are also higher degree relations among the parameters {qij} some of whose in degree 16 follow from the deformed Plücker relation between parameters {pij}: 1 pikpjl = 1 pijpkl + 1 pilpjk + h pijpjkpkl , i < j < k < l. On Some Quadratic Algebras 23 However, we don’t know how to describe the algebra An,h generated by quantum parameters {qij}1≤i<j≤n even for n = 4. The algebra An = gr(An,h) is isomorphic to the quotient algebra of Q[xij , 1 ≤ i < j ≤ n] modulo the ideal generated by the set of relations between “quantum parameters”{ qij := ( 1 zi − zj )2 } 1≤i<j≤n , which correspond to the Dunkl–Gaudin elements {θi}1≤i≤n, see Section 3.2 below for details. In this case the parameters {qij} satisfy the following relations q2 ijq 2 jk + q2 ijq 2 ik + q2 jkq 2 ik = 2qijqikqjk(qij + qjk + qjk) which correspond to the relations (2.8) in the special case h = 0. One can find a set of relations in degrees 6, 7 and 8, namely for a given pair-wise distinct integers 1 ≤ i, j, k, l ≤ n, one has • one relation in degree 6 q2 ijq 2 ikq 2 il + q2 ijq 2 jkq 2 jl + q2 ikq 2 jkq 2 kl + q2 ilq 2 jlq 2 kl − 2qijqikqilqjkqjlqkl ( qij qkl + qkl qij + qik qjl + qjl qik + qil qjk + qjk qil ) + 8qijqikqilqjkqjlqkl = 0; • three relations in degree 7 qik ( qijqilqkl − qijqilqjk + qijqjkqkl − qilqjkqkl )2 = 8q2 ijq 2 ikqjkqkl ( qjk + qjl + qkl ) − 4q2 ijq 2 ilqjl ( q2 jk + q2 kl ) , • one relation in degree 8 q2 ijq 2 ilq 2 jkq 2 kl + q2 ijq 2 ikq 2 jlq 2 kl + q2 ikq 2 ilq 2 jkq 2 jl = 2qijqikqilqjkqjlqkl ( qijqkl + qikqjl + qilqjk ) , However we don’t know does the list of relations displayed above, contains the all independent relations among the elements {qij}1≤i<j≤n in degrees 6, 7 and 8, even for n = 4. In degrees ≥ 9 and n ≥ 5 some independent relations should appear. Notice that the parameters { pij = hqj qi−qj , i < j } satisfy the so-called Gelfand–Varchenko relations, see, e.g., [67] pijpjk = pikpij + pjkpik + hpik, i < j < k, whereas parameters { pij = 1 qi−qj , i < j } satisfy the so-called Arnold relations pijpjk = pikpij + pjkpik, i < j < k. Project 2.9. Find Hilbert series Hilb(An, t) for n ≥ 4.24 24This is a particular case of more general problem we are interested in. Namely, let {fα ∈ R[x1, . . . , xn]}1≤α≤N be a collection of linear forms, and k ≥ 2 be an integer. Denote by I({fα}) the ideal in the ring of polynomials R[z1, . . . , zN ] generated by polynomials Φ(z1, . . . , zN ) such that Φ ( f−k1 , . . . , f−kN ) = 0. Compute the Hilbert series (polynomial?) of the quotient algebra R[z1, . . . , zN ]/I({fα}). 24 A.N. Kirillov For example, Hilb(A3, t) = (1+t)(1+t2) (1−t)2 . Finally, if we set qi := exp(hzi) and take the limit lim h→0 h2qiqj (qi−qj)2 , as a result we obtain the Dunkl–Gaudin parameter qij = 1 (zi−zj)2 . (III) Consider the following representation of the degenerate affine Hecke algebra Hn on the ring of polynomials Pn = Q[x1, . . . , xn]: • the symmetric group Sn acts on Pn by means of operators si = 1 + (xi+1 − xi − h)∂i, i = 1, . . . , n− 1, • yi acts on the ring Pn by multiplication on the variable xi: yi(f(x)) = xif(x), f ∈ Pn. Clearly, yisi − yi+1si = h and yi(si − 1) = (si − 1)yi+1 + xi+1 − xi − h. In the subsequent discussion we will identify the operator of multiplication by the variable xi, namely the operator yi, with xi. This time define uij = pij(si−1), if i < j and set uij = −uji if i > j, where parameters {pij} satisfy the same conditions as in the previous example. Lemma 2.10. The elements {uij , 1 ≤ i < j ≤ n}, satisfy the dynamical classical Yang–Baxter relations displayed in Lemma 2.6, equation (2.7). Therefore, the Dunkl elements θi := xi + ∑ j j 6=i uij , i = 1, . . . , n, form a commutative set of elements. Theorem 2.11 ([54]). Define matrix Mn = (mij)1≤i,j≤n as follows mi,j(u; z1, . . . , zn) =  u− zi + ∑ j 6=i hpij if i = j, −h− pij if i < j, pij if i > j. Then DET ∣∣Mn(u; θ1, . . . , θn) ∣∣ = n∏ j=1 (u− xj). Comments 2.12. Let us list a few more representations of the dynamical classical Yang–Baxter relations. • Trigonometric Calogero–Moser representation. Let i < j, define uij = xj xi − xj (sij − ε), ε = 0 or 1, sij(xi) = xj , sij(xj) = xi, sij(xk) = xk, ∀ k 6= i, j. On Some Quadratic Algebras 25 • Mixed representation: uij = ( λj λi − λj − xj xi − xj ) (sij − ε), ε = 0 or 1, sij(λk) = λk, ∀ k. We set uij = −uji, if i > j. In all cases we define Dunkl elements to be θi = ∑ j 6=i uij . Note that operators rij = ( λi + λj λi − λj − xi + xj xi − xj ) sij satisfy the three term relations: rijrjk = rikrij + rjkrik, and rjkrij = rijrjk + rikrjk, and thus satisfy the classical Yang–Baxter relations. 2.1.2 Step functions and the Dunkl–Uglov representations of the degenerate aff ine Hecke algebras [138] Consider step functions η± : R −→ {0, 1} (Heaviside function) η+(x) = { 1 if x ≥ 0, 0 if x < 0, η−(x) = { 1 if x > 0, 0 if x ≤ 0. For any two real numbers xi and xj set η±ij = η±(xi − xj). Lemma 2.13. The functions ηij satisfy the following relations η±ij + η±ji = 1 + δxi,xj , (η±ij) 2 = η±ij , η±ijη ± jk = η±ikη ± ij + η±jkη ± ik − η ± ik, where δx,y denotes the Kronecker delta function. To introduce the Dunkl–Uglov operators [138] we need a few more definitions and notation. To start with, denote by ∆±i the finite difference operators: ∆±i (f)(x1, . . . , xn) = f(. . . , xi ± 1, . . .). Let as before, {sij , 1 ≤ i 6= j ≤ n, sij = sji}, denotes the set of transpositions in the symmetric group Sn. Recall that sij(xi) = xj , sij(xk) = xk, ∀ k 6= i, j. Finally define Dunkl–Uglov operators d±i : Rn −→ Rn to be d±i = ∆±i + ∑ j<i δxi,xj − ∑ j<i η±jisij + ∑ j>i η±ijsij . To simplify notation, set u±ij := η±ijsij if i < j, and ∆̃±i = ∆±i + ∑ j<i δxi,xj . Lemma 2.14. The operators {u±ij , 1 ≤ i < j ≤ n} satisfy the following relations [ u±ij , u ± ik + u±jk ] + [ u±ik, u ± jk ] + [ u±ik, ∑ j<i δxi,xj ] = 0 if i < j < k. From now on we assume that xi ∈ Z, ∀ i, that is, we will work with the restriction of the all operators defined at beginning of Example 2.28(c), to the subset Zn ⊂ Rn. It is easy to see that under the assumptions xi ∈ Z, ∀ i, we will have ∆±j η ± ij = (η±ij ∓ δxi,xj )∆ ± i . (2.9) Moreover, using relations (2.12), (2.13) one can prove that 26 A.N. Kirillov Lemma 2.15. • [u±ij , ∆̃ ± i + ∆̃±j ] = 0, • [u±ik, ∆̃ ± j ] = [ u±ik, ∑ j<i δxi,xj ] , i < j < k. Corollary 2.16. • The operators {u±ij , 1 ≤ i < j < k ≤ n}, and ∆̃±i , i = 1, . . . , n satisfy the dynamical classical Yang–Baxter relations[ u±ij , u ± ik + u±jk ] + [ u±ik, u ± jk ] + [ u±ik, ∆̃j ] = 0 if i < j < k. • The operators {si := si,i+1, 1 ≤ i < n, and ∆̃±j , 1 ≤ j ≤ n} give rise to two representations of the degenerate affine Hecke algebra Hn. In particular, the Dunkl–Uglov operators are mutually commute: [d±i , d ± j ] = 0 [138]. 2.1.3 Extended Kohno–Drinfeld algebra and Yangian Dunkl–Gaudin elements Definition 2.17. Extended Kohno–Drinfeld algebra is an associative algebra over Q[β] gene- rated by the elements {z1, . . . , zn} and {yij}1≤i 6=j≤n subject to the set of relations (i) The elements {yij{1≤i 6=j≤n satisfy the Kohno–Drinfeld relations • yij = yji, [yij , ykl] = 0 if i, j, k, l are distinct, • [yij , yik + yjk] = 0 = [yij + yik, yjk] if i < j < k. (ii) The elements z1, . . . , zn generate the free associative algebra Fn. (iii) Crossing relations: • [zi, yjk] = 0 if i 6= j, k, [zi, zj ] = β[yij , zi] if i 6= j. To define the (Yangian) Dunkl–Gaudin elements, cf. [54], let us consider a set of elements {pij}1≤i 6=j≤n subject to relations • pij + pji = β, [pij , ykl] = 0 = [pij , zk] for all i, j, k, • pijpjk = pik(pjk − pji) if i < j < k. Let us set uij = pijyij , i 6= j, and define the (Yangian) Dunkl–Gaudin elements as follows θi = zi + ∑ j 6=i uij , i = 1, . . . , n. Proposition 2.18 (cf. [54, Lemma 3.5]). The elements θ1, . . . , θn form a mutually commuting family. Indeed, let i < j, then [θi, θj ] = [zi, zj ] + β[zi, yij ] + pij [yij , zi + zj ] + ∑ k 6=i,j ( pikpjk [ yij + yik, yjk ] + pikpji [ yij , yik + yjk ]) = 0. A representation of the extended Kohno–Drinfeld algebra has been constructed in [54], namely one can take yij := T (1) ij T (1) ji − T (1) jj = yji, zi := βT (2) ii − β 2 T (1) ii ( T (1) ii − 1 ) , On Some Quadratic Algebras 27 pij := βqj qi − qj , i 6= j, where q1, . . . , qn stands for a set of mutually commuting quantum parameters, and { T (s) ij } 1≤i,j≤n s∈Z≥0 denotes the set of generators of the Yangian Y (gln), see, e.g., [106]. A proof that the elements {zi}1≤i≤n and {yij}1≤i 6=j≤n satisfy the extended Kohno–Drinfeld algebra relations is based on the following relations, see, e.g., [54, Section 3],[ T (1) ij , T (s) kl ] = δilT (s) kj − δjkT (s) il , i, j, k, l = 1, . . . , n, s ∈ Z≥0. 2.2 “Compatible” Dunkl elements, Manin matrices and algebras related with weighted complete graphs rKn Let us consider a collection of generators {u(α) ij , 1 ≤ i, j ≤ n, α = 1, . . . , r}, subject to the following relations • either the unitarity (the case of sign “+”) or the symmetry relations (the case of sign “−”)25 u (α) ij ± u (α) ji = 0, ∀α, i, j, (2.10) • local 3-term relations: u (α) ij u (α) jk + u (α) jk u α) ki + u (α) ki u (α) ij = 0, i, j, k are distinct, 1 ≤ α ≤ r. (2.11) We define global 3-term relations algebra 3T (±) n,r as “compatible product” of the local 3-term relations algebras. Namely, we require that the elements U (λ) ij := r∑ α=1 λαu (α) ij , 1 ≤ i, j ≤ n, satisfy the global 3-term relations U (λ) ij U (λ) jk + U (λ) jk U (λ) ki + U (λ) ki U (λ) ij = 0 for all values of parameters {λi ∈ R, 1 ≤ α ≤ r}. It is easy to check that our request is equivalent to a validity of the following sets of relations among the generators { u (α) ij } (a) local 3-term relations: u (α) ij u α) jk + u (α) jk u (α) ki + u α) kiu (α) ij = 0, (b) 6-term crossing relations: u (α) ij u (β) jk + u (β) ij u (α) jk + u (α) k,i u (β) ij u (α) ki + u (α) jk u (β) ki + u (β) jk u (α) ki = 0, i, j, k are distinct, α 6= β. 25More generally one can impose the q-symmetry conditions uij + quji = 0, 1 ≤ i < j ≤ n and ask about relations among the local Dunkl elements to ensure the commutativity of the global ones. As one might expect, the matrix Q := ( θ (a) j ) 1≤a≤r 1≤j≤n composed from the local Dunkl elements should be a q-Manin matrix. See, e.g., [25], or https://en.wikipedia.org/wiki/Manin.matrix for a definition and basic properties of the latter. https://en.wikipedia.org/wiki/Manin.matrix 28 A.N. Kirillov Now let us consider local Dunkl elements θ (α) i := ∑ j 6=i u (α) ij , j = 1, . . . , n, α = 1, . . . , r. It follows from the local 3-term relations (2.11) that for a fixed α ∈ [1, r] the local Dunkl elements{ θ (α) i } 1≤i≤n 1≤α≤r either mutually commute (the sign “+”), or pairwise anticommute (the sign “−”). Similarly, the global 3-term relations imply that the global Dunkl elements θ (λ) i := λ1θ (1) i + · · ·+ λrθ (r) i = ∑ j 6=i U (λ) ij , i = 1, . . . , n, also either mutually commute (the case “+”) or pairwise anticommute (the case “−”). Now we are looking for a set of relations among the local Dunkl elements which is a conse- quence of the commutativity (anticommutativity) of the global Dunkl elements. It is quite clear that if i < j, then [ θ (a) i , θ (b) j ] ± = r∑ a=1 λ2 a [ θ (a) i , θ (a) j ] ± + ∑ 1≤a<b≤r λaλb ([ θ (a) i , θ (b) j ] ± + [ θ (b) i , θ (a) j ] ± ) , and the commutativity (or anticommutativity) of the global Dunkl elements for all (λ1, . . . , λr) ∈ Rr is equivalent to the following set of relations • [θ (a) i , θ (a) j ]± = 0, • [θ (a) i , θ (b) j ]±+[θ (b) i , θ (a) j ]± = 0, a < b and i < j, where by definition we set [a, b]± := ab∓ba. In other words, the matrix Θn := ( θ (a) i ) 1≤a≤r 1≤i≤n should be either a Manin matrix (the case “+”), or its super analogue (the case “−”). Clearly enough that a similar construction can be applied to the algebras studied in Section 2, I–III, and thus it produces some interesting examples of the Manin matrices. It is an interesting problem to describe the algebra generated by the local Dunkl elements { θ (a) i } 1≤a≤r 1≤i≤n and a commutative subalgebra generated by the global Dunkl elements inside the former. It is also an interesting question whether or not the coefficients C1, . . . , Cn of the column characteristic polynomial Detcol \|Θn − tIn\| = n∑ k=0 Ckt n−k of the Manin matrix Θn generate a commutative subalgebra? For a definition of the column determinant of a matrix, see, e.g., [25]. However a close look at this problem and the question posed needs an additional treatment and has been omitted from the content of the present paper. Here we are looking for a “natural conditions” to be imposed on the set of generators {uαij} 1≤α≤r 1≤i,j≤n in order to ensure that the local Dunkl elements satisfy the commutativity (or anticommutativity) relations:[ θ (α) i , θ (β) j ] ± = 0, for all 1 ≤ i < j ≤ n, 1 ≤ α, β ≤ r. The “natural conditions” we have in mind are • locality relations:[ u (α) ij , u (β) kl ] ± = 0, (2.12) On Some Quadratic Algebras 29 • twisted classical Yang–Baxter relations:[ u (α) ij , u (β) jk ] ± + [ u (α) ik , u (β) ji ] ± + [ u (α) ik , u (β) jk ] ± = 0, (2.13) if i, j, k, l are distinct and 1 ≤ α, β ≤ r. Finally we define a multiple analogue of the three term relations algebra, denoted by 3T±(rKn), to be the quotient of the global 3-term relations algebra 3T±n,r modulo the two- sided ideal generated by the left hand sides of relations (2.12), (2.13) and that of the following relations • ( u (α) ij )2 = 0, [ u (α) ij , u (β) ij ] ± = 0, for all i 6= j, α 6= β. The outputs of this construction are • commutative (or anticommutative) quadratic algebra 3T (±)(rKn) generated by the ele- ments { u (α) ij } 1≤i<j≤n α=1,...,r , • a family of nr either mutually commuting (the case “+”), or pair-wise anticommuting (the case “−”) local Dunkl elements { θ (α) i } i=1,...,n α=1,...,r . We expect that the subalgebra generated by local Dunkl elements in the algebra 3T+(rKn) is closely related (isomorphic for r = 2) with the coinvariant algebra of the diagonal action of the symmetric group Sn on the ring of polynomials Q [ X (1) n , . . . , X (r) n ] , where X (j) n stands for the set of variables { x (j) 1 , . . . , x (j) n } . The algebra 3T−(2Kn)anti has been studied in [72] and [12]. In the present paper we state only our old conjecture. Conjecture 2.19 (A.N. Kirillov, 2000). Hilb ( 3T−(3Kn)anti, t ) = (1 + t)n(1 + nt)n−2, where for any algebra A we denote by Aanti the quotient of algebra A by the two-sided ideal generated by the set of anticommutators {ab+ ba \| (a, b) ∈ A×A}. According to observation of M. Haiman [55], the number 2n(n+ 1)n−2 is thought of as being equal to the dimension of the space of triple coinvariants of the symmetric group Sn. 2.3 Miscellany 2.3.1 Non-unitary dynamical classical Yang–Baxter algebra DCYBn Let Ãn be the quotient of the algebra Fn by the two-sided ideal generated by the relations (2.2), (2.5) and (2.6). Consider elements θi = xi + ∑ a6=i uia and θ̄j = −xj + ∑ b6=j ubj , 1 ≤ i < j ≤ n. Clearly, if i < j, then [θi, θ̄j ] + [xi, xj ] = [ n∑ k=1 xk, uij ] + ∑ k 6=i,j wikj , where the elements wijk, i < j, have been defined in Lemma 2.2, equation (2.3). Therefore the elements θi and θ̄j commute in the algebra Ãn. 30 A.N. Kirillov In the case when xi = 0 for all i = 1, . . . , n, the relations wijk := [uij , uik + ujk] + [uik, ujk] = 0 if i, j, k are all distinct, are well-known as the non-unitary classical Yang–Baxter relations. Note that for a given triple of pair-wise distinct (i, j, k) one has in fact 6 relations. These six relations imply that [θi, θ̄j ] = 0. However, in general, [θi, θj ] = [∑ k 6=i,j uik, uij + uji ] 6= 0. Dynamical classical Yang–Baxter algebra DCYBn. In order to ensure the commuta- tivity relations among the Dunkl elements (2.1), i.e., [θi, θj ] = 0 for all i, j, let us remark that if i 6= j, then [θ,θj ] = [xi + uij , xj + uji] + [xi + xj , uij ] + [ uij , n∑ k=1 xk ] + n∑ k=1 k 6=i,j [uij + uik, ujk] + [uik, uji] + [xi, ujk] + [uik, xj ] + [xk, uij ]. Definition 2.20. Define dynamical non-unitary classical Yang–Baxter algebra DNUCYBn to be the quotient of the free associative algebra Q〈{xi, 1 ≤ i ≤ n}, {uij}1≤i 6=j≤n〉 by the two-sided ideal generated by the following set of relations • zero curvature conditions: [xi + uij , xj + uji] = 0, 1 ≤ i 6= j ≤ n, (2.14) • conservation laws conditions:[ uij , n∑ k=1 xk ] = 0 for all i 6= j, k. • crossing relations: [xi + xj , uij ] = 0, i 6= j. • twisted dynamical classical Yang–Baxter relations: [uij + uik, ujk] + [uik, uji] + [xi, ujk] + [uik, xj ] + [xk, uij ] = 0, i, j, k are distinct. It is easy to see that the twisted classical Yang–Baxter relations [uij + uik, ujk] + [uik, uji] = 0, i, j, k are distinct, (2.15) for a fixed triple of distinct indices i, j, k contain in fact 3 different relations whereas the non-unitary classical Yang–Baxter relations [uij + uik, ujk] + [uij , uik], i, j, k are distinct, contain 6 different relations for a fixed triple of distinct indices i, j, k. On Some Quadratic Algebras 31 Definition 2.21. • Define dynamical classical Yang–Baxter algebra DCYBn to be the quotient of the algebra DNUCYBn by the two-sided ideal generated by the elements∑ k 6=i,j [uik, uij + uji] for all i 6= j. • Define classical Yang–Baxter algebra CYBn to be the quotient of the dynamical classical Yang–Baxter algebra DCYBn by the set of relations xi = 0 for i = 1, . . . , n. Example 2.22. Define pij(z1, . . . , zn) =  zi zi − zj if 1 ≤ i < j ≤ n, − zj zj − zi if n ≥ i > j ≥ 1. Clearly, pij+pji = 1. Now define operators uij = pijsij , and the truncated Dunkl operators to be θi = ∑ j 6=i uij , i = 1, . . . , n. All these operators act on the field of rational functions Q(z1, . . . , zn); the operator sij = sji acts as the exchange operator, namely, sij(zi) = zj , sij(zk) = zk, ∀ k 6= i, j, sij(zj) = zi. Note that this time one has p12p23 = p13p12 + p23p13 − p13. It is easy to see that the operators {uij , 1 ≤ i 6= j ≤ n} satisfy relations (3.1), and therefore, satisfy the twisted classical Yang–Baxter relations (2.13). As a corollary we obtain that the truncated Dunkl operators {θi, i = 1, . . . , n} are pair-wise commute. Now consider the Dunkl operator Di = ∂zi + hθi, i = 1, . . . , n, where h is a parameter. Clearly that [∂zi + ∂zj , uij ] = 0, and therefore [Di, Dj ] = 0, ∀ i, j. It easy to see that si,i+1Di −Di+1si,i+1 = h, [Di, sj,j+1] = 0 if j 6= i, i+ 1. In such a manner we come to the well-known representation of the degenerate affine Hecke algebra Hn. 2.3.2 Dunkl and Knizhnik–Zamolodchikov elements Assume that ∀ i, xi = 0, and generators {uij , 1 ≤ i < j ≤ n} satisfy the locality conditions (2.2) and the classical Yang–Baxter relations [uij , uik + ujk] + [uik, ujk] = 0 if 1 ≤ i < j < k ≤ n. Let y, z, t1, . . . , tn be parameters, consider the rational function FCYB(z; t) := FCYB(z; t1, . . . , tn) = ∑ 1≤i<j≤n (ti − tj)uij (z − ti)(z − tj) . Then [FCYB(z; t), FCYB(y; t)] = 0 and Resz=ti FCYB(z; t) = θi. 32 A.N. Kirillov Now assume that a set of generators {cij , 1 ≤ i 6= j ≤ n} satisfy the locality and symmetry (i.e., cij = cji) conditions, and the Kohno–Drinfeld relations: [cij , ckl] = 0 if {i, j} ∩ {k, l} = ∅, [cij , cjk + cik] = 0 = [cij + cik, cjk], i < j < k. Let y, z, t1, . . . , tn be parameters, consider the rational function FKD(z; t) := FKD(z; t1, . . . , tn) = ∑ 1≤i 6=j≤n cij (z − ti)(ti − tj) = ∑ 1≤i<j≤n cij (z − ti)(z − tj) . Then [FKD(z; t), FKD(y; t)] = 0 and Resz=ti FKD(z; t) = KZi, where KZi = n∑ j=1 j 6=i cij ti − tj denotes the truncated Knizhnik–Zamolodchikov element. 2.3.3 Dunkl and Gaudin operators (a) Rational Dunkl operators. Consider the quotient of the algebra DCYBn, see Defini- tion 2.3, by the two-sided ideal generated by elements {[xi + xj , uij ]} and {[xk, uij ], k 6= i, j}. Clearly the Dunkl elements (2.1) mutually commute. Now let us consider the so-called Calogero– Moser representation of the algebra DCYBn on the ring of polynomials Rn := R[z1, . . . , zn] given by xi(p(z)) = λ ∂p(z) ∂zi , uij(p(z)) = 1 zi − zj (1− sij)p(z), p(z) ∈ Rn. The symmetric group Sn acts on the ring Rn by means of transpositions sij ∈ Sn: sij(zi) = zj , sij(zj) = zi, sij(zk) = zk if k 6= i, j. In the Calogero–Moser representation the Dunkl elements θi becomes the rational Dunkl operators [35], see Definition 1.1. Moreover, one has [xk, uij ] = 0 ifk 6= i, j, and xiuij = uijxj + 1 zi − zj (xi − xj − uij), xjuij = uijxi − 1 zi − zj (xi − xj − uij). (b) Gaudin operators. The Dunkl–Gaudin representation of the algebra DCYBn is defined on the field of rational functions Kn := R(q1, . . . , qn) and given by xi(f(q)) := λ ∂f(q) ∂qi , uij = sij qi − qj , f(q) ∈ Kn, but this time we assume that w(qi) = qi, ∀ i ∈ [1, n] and for all w ∈ Sn. In the Dunkl–Gaudin representation the Dunkl elements becomes the rational Gaudin operators, see, e.g., [108]. More- over, one has [xk, uij ] = 0, if k 6= i, j, and xiuij = uijxj − uij qi − qj , xjuij = uijxi + uij qi − qj . On Some Quadratic Algebras 33 Comments 2.23. It is easy to check that if f ∈ R[z1, . . . , zn], and xi := ∂ ∂zi , then the following commutation relations are true xif = fxi + ∂ ∂zi (f), uijf = sij(f)uij + ∂zi,zj (f). Using these relations it easy to check that in the both cases (a) and (b) the elementary symmetric polynomials ek(x1, . . . , xn) commute with the all generators {uij}1≤i,j≤n, and therefore commute with the all Dunkl elements {θi}1≤i≤n. Let us stress that [θi, xk] 6= 0 for all 1 ≤ i, k ≤ n. Project 2.24. Describe a commutative algebra generated by the Dunkl elements {θi}1≤i≤n and the elementary symmetric polynomials {ek(x1, . . . , xn)}1≤k≤n. 2.3.4 Representation of the algebra 3Tn on the free algebra Z〈t1, . . . , tn〉 Let Fn = Z〈t1, . . . , tn〉 be free associative algebra over the ring of integers Z, equipped with the action of the symmetric group Sn: sij(ti) = tj , sij(tk) = tk, ∀ k 6= i, j. Define the action of uij ∈ 3Tn on the set of generators of the algebra Fn as follows uij(tk) = δi,ktitj − δj,ktjti. The action of generator uij on the whole algebra Fn is defined by linearity and the twisted Leibniz rule: uij(1) = 0, uij(a+ b) = uij(a) + uij(b), uij(ab) = uij(a)b+ sij(a)uij(b). It is easy to see from (2.14) that sijujk = uiksij , sijukl = uklsij if {i, j} ∩ {k, l} = ∅, uij + uji = 0. Now let us consider operator uijk := uijujk − ujkuik − uikuij , 1 ≤ i < j < k ≤ n. Lemma 2.25. uijk(ab) = uijk(a)b+ sijsjk(a)uijk(b), a, b ∈ Fn. Lemma 2.26. uijk(a) = 0 ∀ a ∈ Fn. Indeed, uijk(ti) = −ujk(uij(ti))− uik(uij(ti)) = −tiujk(tk)− uik(ti)tj = ti(tktj)− (titk)tj = 0, uijk(tk) = uij(ujk(tk))− ujk(uik(tk)) = −uij(tktj) + ujk(tkti) = tk(uij(tj) + ujk(tk)ti = 0, uijk(tj) = uij(ujk(tj))− uik(uij(tj)) = −uij(tj)tk − tjuik(ti) = (tjti)tk − tj(titk) = 0. Therefore Lemma 2.26 follows from Lemma 2.25. Let F•n be the quotient of the free algebra Fn by the two-sided ideal generated by elements t2i tj − tjt2i , 1 ≤ i 6= j ≤ n. Since u2 i,j(ti) = tit 2 j − t2j ti, one can define a representation of the algebra 3T (0) n on that F•n. One can also define a representation of the algebra 3T (0) n on that F (0) n , where F (0) n denotes the quotient of the algebra Fn by the two-sided ideal generated by elements {t2i , 1 ≤ i ≤ n}. Note that (ui,kuj,kui,j)(tk) = [titjti, tk] 6= 0 in the algebra F (0) n , but the 34 A.N. Kirillov elements ui,jui,kuj,kui,j , 1 ≤ i < j < k ≤ n, which belong to the kernel of the Calogero–Moser representation [72], act trivially both on the algebras F (0) n and that F•n. Note finally that the algebra F (0) n is Koszul and has Hilbert series Hilb ( F (0) n , t ) = 1+t 1−(n−1)t , whereas the algebra F•n is not Koszul for n ≥ 3, and Hilb(F•n, t) = 1 (1− t)(1− (n− 1)t)(1− t2)n−1 . In Appendix A.5 we apply the representation introduced in this section to the study of relations in the subalgebra Z (0) n of the algebra 3T (0) n generated by the elements u1,n, . . . , un−1,n. To distinguish the generators {uij} of the algebra 3T (0) n from the introduced in this section operators uij acting on it, in Appendix A.5 we will use for the latter notation ∇ij := uij . 2.3.5 Kernel of Bruhat representation Bruhat representations, classical and quantum, of algebras 3T (0) n and 3QTn can be seen as a con- necting link between commutative subalgebras generating by either additive or multiplicative Dunkl elements in these algebras, and classical and quantum Schubert and Grothendieck calculi. (Ia) Bruhat representation of algebra 3T (0) n , cf. [45]. Define action of ui,j ∈ 3T (0) n on the group ring of the symmetric group Z[Sn] as follows: let w ∈ Sn, then ui,jw = { wsij if l(wsij) = l(w) + 1, 0 otherwise. Let us remind that sij ∈ Sn denotes the transposition that interchanges i and j and fixes each k 6= i, j; for each permutation u ∈ Sn, l(u) denotes its length. (Ib) Quantum Bruhat representation of algebra 3QTn, cf. [45]. Let us remind that algebra 3QTn is the quotient of the 3-term relations algebra 3Tn by the two-sided ideal generated by the elements {u2 ij , \|j − i\| ≥ 2} ⋃ {u2 i,i+1 = qi, i = 1, . . . , n− 1}. Define the Z[q]−linear action of ui,j ∈ 3QTn, i < j, on the extended group ring of the symmetric group Z[q][Sn] as follows: let w ∈ Sn, and qij = qiqi+1 · · · qj−1, i < j, then ui,jw =  wsij if l(wsij) = l(w) + 1, qijwsij if l(wsij) = l(w)− l(sij), 0 otherwise. Let us remind, see, e.g., [92], that in general one has l(wsij) = { l(w)− 2eij − 1 if w(i) > w(j), l(w) + 2 eij + 1 if w(i) < w(j). Here eij(w) denotes the number of k such that i < k < j and w(k) lies between w(i) and w(j). In particular, l(wsij) = l(w) + 1 iff eij(w) = 0 and w(i) < w(j); l(wsij) = l(w) − l(sij) = l(w)− 2(j − i) + 1 iff w(i) > w(j) and eij = j − i− 1 is the maximal possible. (II) Kernel of the Bruhat representation. It is not difficult to see that the following elements of degree three and four belong to the kernel of the Bruhat representation: (IIa) ui,jui,kui,j and ui,kuj,kui,k if 1 ≤ i < j < k ≤ n; On Some Quadratic Algebras 35 (IIb) ui,kui,luj,l and uj,lui,lui,k; (IIc) uiluikujluil, uiluijukluil, uikuilujkuik, uijuikuiluij , uikuiluijuik if 1 ≤ i < j < k < l ≤ n. This observation motivates the following definition. Definition 2.27. The reduced 3-term relation algebra 3T red n is defined to be the quotient of the algebra 3T (0) n by the two-sided ideal generated by the elements displayed in IIa–IIc above. Example 2.28. Hilb ( 3T red 3 , t ) = (1, 3, 4, 1), dim ( 3T red 3 ) = 9, Hilb ( 3T red 4 , t ) = (1, 6, 19, 32, 19, 6, 1), dim ( 3T red 4 ) = 84, Hilb ( 3T red 5 , t ) = (1, 10, 55, 190, 383, 370, 227, 102, 34, 8, 1), dim ( 3T red 5 ) = 1374. We expect that dim(3T redn )(n2)−1 = 2(n− 1) if n ≥ 3. Theorem 2.29. 1. The algebra 3T red n is finite-dimensional, and its Hilbert polynomial has degree ( n 2 ) . 2. The maximal degree ( n 2 ) component of the algebra 3T red n has dimension one and generated by any element which is equal to the product (in any order) of all generators of the algebra 3T red n . 3. The subalgebra in 3T red n generated by the elements {ui,i+1, i = 1, . . . , n− 1} is canonically isomorphic to the nil-Coxeter algebra NCn. In particular, its Hilbert polynomial is equal to [n]t! := n∏ j=1 (1−tj) 1−t , and the element n−1∏ j=1 1∏ a=j ua,a+1 of degree ( n 2 ) generates the maximal degree component of the algebra 3T red n . 4. The subalgebra over Z generated by the Dunkl elements {θ1, . . . , θn} in the algebra 3T red n is canonically isomorphic to the cohomology ring H∗(F ln,Z) of the type A flag variety F ln. A definition of the nil-Coxeter algebra NCn one can find in Section 4.1.1. It is known, see [8] or Section 4.1.1, that the subalgebra generated by the elements {ui,i+1, i = 1, . . . , n− 1} in the whole algebra 3T (0) n is canonically isomorphic to the nil-Coxeter algebra NCn as well. We expect that the kernel of the Bruhat representation of the algebra 3T (0) n is generated by all monomials of the form ui1,j1 · · ·uik,jk such that the sequence of transpositions ti1,j1 , . . . , tik,jk does not correspond to a path in the Bruhat graph of the symmetric group Sn. For example if 1 ≤ i < j < k < l ≤ n, the elements ui,kui,luj,l and uj,lui,lui,k do belong to the kernel of the Bruhat representation. Problem 2.30. 1. The image of the Bruhat representation of the algebra 3T (0) n defines a subalgebra Im ( 3T (0) n ) ⊂ EndQ(Q[Sn]). Does this image isomorphic to the algebra 3T red n ? Compute Hilbert polynomials of algebras Im ( 3T (0) n ) and 3T red n . 2. Describe the image(s) of the affine nil-Coxeter algebra ÑCn, see Section 4.1.1, in the algebras 3T red n and EndQ(Q[Sn]). 36 A.N. Kirillov 2.3.6 The Fulton universal ring [47], multiparameter quantum cohomology of flag varieties [45] and the full Kostant–Toda lattice [29, 80] Let Xn = (x1, . . . , xn) be be a set of variables, and g := g(n) = {ga[b] \| a ≥ 1, b ≥ 1, a+ b ≤ n} be a set of parameters; we put deg(xi) = 1 and deg(ga[b]) = b + 1, and set gk[0] := xk, k = 1, . . . , n. For a subset S ⊂ [1, n] we denote by XS the set of variables {xi \| i ∈ S}. Let t be an auxiliary variable, denote by M = (mij)1≤i,j≤n the matrix of size n by n with the following elements: mi,j =  xi + t if i = j, gi[j − i] if j > i, −1 if i− j = 1, 0 if i− j > 1. Let Pn(Xn, t) = det \|M \|. Definition 2.31. The Fulton universal ring Rn−1 is defined to be the quotient26 Rn−1 = Z [ g(n) ] [x1, . . . , xn]/〈Pn(Xn, t)− tn〉. Lemma 2.32. Let Pn(Xn, t) = n∑ k=0 ck(n)tn−k, c0(n) = 1. Then ck(n) := ck ( n;Xn, g (n) ) = ∑ 1≤i1<i2<···<is<n j1≥1,...,js≥1 m:= ∑ (ja+1)≤n s∏ a=1 gia [ja]ek−m ( X [1,n]\ s⋃ a=1 [ia,ia+ja] ) , (2.16) where in the summation we assume additionally that the sets [ia, ia+ja] := {ia, ia+1, . . . , ia+ja}, a = 1, . . . , s, are pair-wise disjoint. It is clear that Rn−1 = Z[g(n)][x1, . . . , xn]/〈cn(1), . . . , cn(n)〉. One can easily see that the coefficients ck(n) and gm[k] satisfy the following recurrence relations [47]: ck(n) = ck(n− 1) + k−1∑ a=0 gn−a[a]ck−a−1(n− a− 1), c0(n) = 1, gm[k] = ck+1(m+ k)− ck+1(m+ k − 1)− k−1∑ a=0 gm+k−a[a]ck−a(m+ k − a), gm[0] := xm. On the other hand, let {qij}1≤i<j≤n be a set of (quantum) parameters, and e (q) k (Xn) be the multiparameter quantum elementary polynomial introduced in [45]. We are interested in to describe a set of relations between the parameters {gi[j]} i≥1,j≥1 i+j≤n and the quantum parameters {qij}1≤i<j≤n which implies that ck(n) = e (q) k (Xn) for k = 1, . . . , n. 26If P (t,Xn) = ∑ k≥1 fk(Xn)tk, fk(Xn) ∈ Q[Xn] is a polynomial, we denote by 〈P (t,Xn)〉 the ideal in the polynomial ring Q[Xn] generated by the coefficients {f1, f2, . . .}. On Some Quadratic Algebras 37 To start with, let us recall the recurrence relations among the quantum elementary polynomials, cf. [117]. To do so, consider the generating function En ( Xn; {qij}1≤i<j≤n ) = n∑ k=0 e (q) k (Xn)tn−k. Lemma 2.33 ([41, 117]). One has En ( Xn; {qij}1≤i<j≤n ) = (t+ xn)En−1 ( Xn−1; {qij}1≤i<j≤n−1 ) + n−1∑ j=1 qjnEn−2 ( X[1,n−1]\{j}; {qa,b} 1≤a<b≤n−1 a6=j,b 6=j ) . Proposition 2.34. Parameters {ga[b]} can be expressed polynomially in terms of quantum pa- rameters {qij} and variables x1, . . . , xn, in a such way that ck(n) = e (q) k (Xn), ∀ k, n. Moreover, • ga[b] = a∑ k=1 qk,a+b a+b−1∏ j=a+1 (xj − xk) + lower degree polynomials in x1, . . . , xn, • the quantum parameters {qij} can be presented as rational functions in terms of variables x1, . . . , xn and polynomially in terms of parameters {ga[b]} such that the equality ck(n) = e (q) k (Xn) holds for all k, n. In other words, the transformation {qij}1≤i<j≤n ←→ {ga[b]} a+b≤n a≥1, b≥1 defines a “birational transformation” between the algebra Z[g(n)][Xn]/〈Pn(Xn, t)− tn〉 and mul- tiparameter quantum deformation of the algebra H∗(F ln,Z). Example 2.35. Clearly, gn−1[1] = n−1∑ j=1 qj,n, n ≥ 2 and gn−2[2] = n−2∑ j=1 qjn(xn−1 − xj), n ≥ 3. Moreover g1[3] = q14 ( (x2 − x1)(x3 − x1) + q23 − q12 ) + q24 ( q13 − q12 ) , g2[3] = q15 ( (x3 − x1)(x4 − x1) + q24 + q34 − q12 − q13 ) + q25 ( (x3 − x2)(x4 − x2) + q14 + q34 − q12 − q23 ) + q35 ( q14 + q24 − q13 − q23 ) . Comments 2.36. The full Kostant–Toda lattice (FKTL for short) has been introduced in the end of 70′s of the last century by B. Kostant and since that time has been extensively studied both in Mathematical and Physical literature. We refer the reader to the original paper by B. Kostant [29, 80] for the definition of the FKTL and its basic properties. In the present paper we just want to point out on a connection of the Fulton universal ring and hence the multiparameter deformation of the cohomology ring of complete flag varieties, and polynomial integral of motion of the FKTL. Namely, Polynomials ck(n;Xn, g (n)) defined by (2.16) coincide with the polynomial integrals of motion of the FKTL. It seems an interesting task to clarify a meaning of the FKTL rational integrals of motion in the context of the universal Schubert calculus [47] and the algebra 3HTn(0), as well as any meaning of universal Schubert or Grothendieck polynomials in the context of the Toda or full Kostant–Toda lattices. 38 A.N. Kirillov 3 Algebra 3HTn Consider the twisted classical Yang–Baxter relation [uij + uik, ujk] + [uik, uji] = 0, where i, j, k are distinct. Having in mind applications of the Dunkl elements to combinatorics and algebraic geometry, we split the above relation into two relations uijujk = ujkuik − uikuji and ujkuij = uikujk − ujiuik (3.1) and impose the following unitarity constraints uij + uji = β, where β is a central element. Summarizing, we come to the following definition. Definition 3.1. Define algebra 3Tn(β) to be the quotient of the free associative algebra Z[β]〈uij , 1 ≤ i < j ≤ n〉 by the set of relations • locality: uijukl = ukluij if {i, j} ∩ {k, l} = ∅, • 3-term relations: uijujk = uikuij + ujkuik − βuik, and ujkuij = uijuik + uikujk − βuik if 1 ≤ i < j < k ≤ n. It is clear that the elements {uij , ujk, uik, 1 ≤ i < j < k ≤ n} satisfy the classical Yang– Baxter relations, and therefore, the elements { θi := ∑ j 6=i uij , 1 = 1, . . . , n } form a mutually commuting set of elements in the algebra 3Tn(β). Definition 3.2. We will call θ1, . . . , θn by the (universal) additive Dunkl elements. For each pair of indices i < j, we define element qij := u2 ij − βuij ∈ 3Tn(β). Lemma 3.3. 1. The elements {qij , 1 ≤ i < j ≤ n} satisfy the Kohno–Drinfeld relations (known also as the horizontal four term relations) qijqkl = qklqij if {i, j} ∩ {k, l} = ∅, [qij , qik + qjk] = 0, [qij + qik, qjk] = 0 if i < j < k. 2. For a triple (i < j < k) define uijk := uij − uik + ujk. Then u2 ijk = βuijk + qij + qik + qjk. 3. Deviation from the Yang–Baxter and Coxeter relations: uijuikujk − ujkuikuij = [uik, qij ] = [qjk, uik], uijujkuij − ujkuijujk = qijuik − uikqjk. On Some Quadratic Algebras 39 Comments 3.4. It is easy to see that the horizontal 4-term relations listed in Lemma 3.3(1), are consequences of the locality conditions among the generators {qij}, together with the com- mutativity conditions among the Jucys–Murphy elements di := n∑ j=i+1 qij , i = 2, . . . , n, namely, [di, dj ] = 0. In [72] we describe some properties of a commutative subalgebra generated by the Jucys–Murphy elements in the (nil27) Kohno–Drinfeld algebra. It is well-known that the Jucys–Murphy elements generate a maximal commutative subalgebra in the group ring of the symmetric group Sn. It is an open problem describe defining relations among the Jucys–Murphy ele- ments in the group ring Z[Sn]. Finally we introduce the “Hecke quotient” of the algebra 3Tn(β), denoted by 3HTn(β). Definition 3.5. Define algebra 3HTn(β) to be the quotient of the algebra 3Tn(β) by the set of relations qijqkl = qklqij for all i, j, k, l. In other words we assume that the all elements {qij , 1 ≤ i < j ≤ n} are central in the algebra 3Tn(β). From Lemma 3.3 follows immediately that in the algebra 3HTn(β) the elements {uij} satisfy the multiplicative (or quantum) Yang–Baxter relations uijuikujk = ujkuikuij if i < j < k. (3.2) To underline the dependence of the algebra 3HTn(β) on the central elements q := {qij}, we will use for the former the notation 3T (q) n (β) as well. Exercises 3.6 (some relations in the algebra 3T (q) n (β)). 1. Noncommutative analogue of recurrence relation among the Catalan numbers [70, 72], cf. Section 5.1. Let k, n be positive integers, k < n and i1, . . . , ik, 1 ≤ ik < n, be a collection of pairwise distinct integers. Prove the following identity in the algebra 3T (q) n (β)28 k∏ a=1 uia,ia+1 + k+1∑ r=2 ( n∏ a=r β(uia,ia+1)uiak+1 ,ia1 ( r−2∏ a=1 uia,ia+1 )) = β k∏ a=1 uia,ia+1 − β ( ui1,ik+1 RI\{ik+1} −RI\{ik+1}uik,ik+1 ) , where RI denotes the r.h.s. of the above identity. For example, 12 23 + 23 31 + 31 12 = β(12− 13 + 23), 12 23 34 + 23 34 41 + 34 41 12 + 41 12 23 = β(12 23− 14(12− 13 + 23) + (12− 13 + 23)34), 27That is the quotient of the Kohno–Drinfeld algebra generated by the elements {qij} by the two-sided ideal generated by the elements {q2 ij}1≤i,j≤n. 28Hint: denote the r.h.s. of of the identity stated in item (1) by RI . One possible proof is based on induction and examination of the element RI∪{ik+2} := uia1,ia+2 RI −RIuia+1,iai+2 . 40 A.N. Kirillov where we use short notation ij := uij . See Introduction, summation formula, A, for an interpretation of the above formula in the case β = 0, qij = 0, ∀ i, j. Note that the above formula does not depend on deformation (or quantum) parameters {qij}, in particular it also true for the algebras 3T (Γ) associated with a simple graph Γ, and gives rise to quantum as well as K-theoretic deformations of the Orlik–Terao algebra of a simple graph, cf. [89]. 2. Cyclic relations, cf. [45]. Let i1, i2, . . . , ik, 1 ≤ ia ≤ n be a collection of pairwise distinct integers. Show that k−1∑ r=1 ( k∏ a=r+1 ui1,ia )( r∏ a=2 ui1,ia ) uir+1,i1 = − ( k∑ a=2 qi1,ia ( k∏ b=a+1 uia,ib )( a−1∏ b=2 uia,ib )) . For example, 12 13 14 21 + 13 14 12 31 + 14 12 13 41 = −q12 23 24− q13 34 32− q14 42 43. Note that the r.h.s. does not depend on parameter β. 3.1 Modified three term relations algebra 3MTn(β, ψ) Let β, {qij = qji, ψij = ψji, 1 ≤ i, j ≤ n}, be a set of mutually commuting elements. Definition 3.7. Modified 3-term relation algebra 3MTn(β, q, ψ) is an associative algebra over the ring of polynomials Z[β, qij , ψij ] with the set of generators {uij , 1 ≤ i, j ≤ n} subject to the set of relations • uij + uji = β, uijukl = ukluij if {i, j} ∩ {k, l} = ∅, • three term relations: uijujk + ukiuij + ujkuki = β(uij + uik + ujk) if i, j, k are distinct, • u2 ij = βuuj + qij + ψij if i 6= j, • uijψkl = ψkluij if {i, j} ∩ {k, l} = ∅, • exchange relations: uijψjk = ψikuij if i, j, k are distinct, • elements β, {qij , 1 ≤ i, j ≤ n} are central. It is easy to see that in the algebra 3MTn(β, q,ψ) the generators {uij} satisfy the modified Coxeter and modified quantum Yang–Baxter relations, namely • modified Coxeter relations: uijujkuij − ujkuijujk = (qij − qjk)uik, • modified quantum Yang–Baxter relations: uijuikujk − ujkuikuij = (ψjk − ψij)uik if i, j, k are distinct. Clearly the additive Dunkl elements { θi := ∑ j 6=i uij , i = 1, . . . , n } generate a commutative subalgebra in 3MTn(β, ψ). It is still possible to describe relations among the additive Dunkl elements [72], cf. [74]. However we don’t know any geometric interpretation of the commutative algebra obtained. It is not unlikely that this commutative subalgebra is a common generalization of the small quantum cohomology and elliptic cohomology (remains to be defined!) of complete flag varieties. The algebra 3MTn(β = 0, q = 0, ψ) has an elliptic representation [72, 74]. Namely, uij := σλi−λj (zi − zj)sij , qij = ℘(λi − λj), ψij = −℘(zi − zj), On Some Quadratic Algebras 41 where {λi, i = 1, . . . , n} is a set of parameters (e.g., complex numbers), and {z1, . . . , zn} is a set of variables; sij , i < j, denotes the transposition that swaps i on j and fixes all other variables; σλ(z) := θ(z − λ)θ′(0) θ(z)θ(λ) denotes the Kronecker sigma function; ℘(z) denotes the Weierstrass P -function. “Multiplicative” version of the elliptic representation. Let q be parameter. In this place we will use the same symbol θ(x) to denote the “multiplicative” version of the Riemann theta function θ(x) := θ(x; q) = (x; q)∞(q/x; q)∞, where by definition (x; q)∞ = (x)∞ = ∏ k≥0 (1 − x qk). Let us state some well-known properties of the Riemann theta function: • θ(qx; q) = θ(1/x; q) = −x−1θ(x; q), • functional equation: x/yθ ( ux±1 ) θ ( yv±1 ) + θ ( uv±1 ) θ ( xy±1 ) = θ ( uy±1 ) θ ( xv±1 ) , where by definition θ(xy±1) := θ(xy)θ(xy−1). • Jacobi triple product identity: (q; q)∞θ(x; q) = ∑ n∈Z (−x)nq( n 2). One can easily check that after the change of variables x := ( z2 λw )1/2 , y := (w λ )1/2 , u := ( w λµ2 )1/2 , v := (wλ)1/2, the functional equation for the Riemann theta function θ(x) takes the following form σλ(z) σµ(w) = σλµ(z)σµ(w/z) + σλµ(w)σλ(z/w), where σλ(z) := θ(z/λ) θ(z)θ ( λ−1 ) denotes the (multiplicative) Kronecker sigma function. Therefore, the operators uij(f) := σλi/λj (zi/zj)sij(f), where sij denotes the exchange operator which swaps the variables zi and zj , namely sij(zi) = zj , sij(zj) = zi, sij(zk) = zk, ∀ k 6= i, j, and sij acts trivially on dynamical parameters λi, namely, sij(λk) = λk, ∀ k, give rise to a representation of the algebra 3MTn(β = 0, q = 0, ψ). The 3-term relations among the elements {uij} are consequence (in fact equivalent) to the famous Jacobi–Riemann 3-term relation of degree 4 among the theta function θ(z), see, e.g., [141, p. 451, Example 5]. In several cases, see Introduction, relations (A) and (B), identities among the Riemann theta functions can be rewritten in terms of the elliptic Kronecker sigma functions 42 A.N. Kirillov and turn out to be a consequence of certain relations in the algebra 3MTn(β = 0, q = 0, ψ) for some integer n, and vice versa29. The algebra 3HTn(β) is the quotient of algebra 3MTn(β, q, ψ) by the two-sided ideal gene- rated by the elements {ψij}. Therefore the elements {uij} of the algebra 3HTn(β) satisfy the quantum Yang–Baxter relations uijuikujk = ujkuikuij , i < j < k, and as a consequence, the multiplicative Dunkl elements Θi = 1∏ a=i−1 (1 + hua,i) −1 n∏ a=i+1 (1 + hui,a), i = 1, . . . , n, u0,i = ui,n+1 = 0 generate a commutative subalgebra in the algebra 3HTn(β), see Section 3.1. We emphasize that the Dunkl elements Θj , j = 1, . . . , n, do not pairwise commute in the algebra 3MTn(β, q,ψ), if ψij 6= 0 for some i 6= j. One way to construct a multiplicative analog of additive Dunkl elements θi := ∑ j 6=i uij is to add a new set of mutually commuting generators denoted by {ρij , ρij +ρji = 0, 1 ≤ i 6= j ≤ n} subject to the crossing relations • ρij commutes with β, qkl and ψk,l for all i, j, k, l, • ρijukl = uklρij if {i, j} ∩ {k, l} = ∅, ρijujk = ujkρik if i, j, k are distinct, • ρ2 ij − βρij + ψij = ρ2 jk − βρjk + ψjk for all triples 1 ≤ i < j < k ≤ n. Under these assumptions one can check that elements Rij := ρij + uij , 1 ≤ i < j ≤ n satisfy the quantum Yang–Baxter relations RijRikRjk = RjkRikRij , i < j < k. In the case of elliptic representation defined above, one can take ρij := σµ(zi − zj), where µ ∈ C∗ is a parameter. This solution to the quantum Yang–Baxter equation has been discovered in [130]. It can be seen as an operator form of the famous (finite-dimensional) solution to QYBE due to A. Belavin and V. Drinfeld [9]. One can go to one step more and add to the algebra in question a new set of generators corresponding to the shift operators Ti,q : zi −→ qzi, cf. [40]. In this case one can define multiplicative Dunkl elements which are closely related with the elliptic Ruijsenaars–Schneider–Macdonald operators. 3.1.1 Equivariant modified three term relations algebra Let h = (h2, . . . , hn) be a set of parameters. We define equivariant modified 3-term relations al- gebra 3EMTn(β, h, q, ψ) to be the extension of the algebra 3TMn(β, q, ψ) by the set of mutually commuting generators {y1, . . . , yn} subject to the crossing relations • yiujk = ujkyi if i 6= j, k, yiuij = uijyj + hj , yjuij = uijyi − hj , i < j, • [yk, qij ] = 0 = [yk, ψij ] for all i, j, k. 29It is commonly believed that any identity between the Riemann theta functions is a consequence of the Jacobi– Riemann three term relations among the former. However we do not expect that the all hypergeometric type identities among the Riemann theta functions can be obtained from certain relations in the algebra 3MTn(β = 0, q = 0, ψ) after applying the elliptic representation of the latter. On Some Quadratic Algebras 43 It is clear that the additive Dunkl elements θi = yi + ∑ j 6=i uij , i = 1, . . . , n, are pair-wise commute. For simplicity’s sake, we shall restrict our consideration to the case β = 0. Theorem 3.8 (generalized Pieri’s rule, cf. [72, 74, 117]). Let 1 ≤ m ≤ n, then e (h,q) k ( θ (n) 1 , . . . , θ(n) m ) := ∑ A⊂[1,m], \|A\|=2r B⊂[1,m]\A,\|B\|=2s Hr(A)MB({qij})ek−2r−2s(Θ[1,m]\(A∪B)) = ∑ A⊂[1,m] YA ∑ B⊂[1,m]\A \|B\|=2s (−1)sMB({ψij}) ∑ I⊂[1,n]\A, I∩B=∅ \|A\|+\|B\|+\|I\|=k ∏ (iα,jα)∈I×I 1≤iα≤m<jα≤n, ∀α i1,...,iI are distinct uiα,jα , where for any subset C ⊂ [1, n] we put YC := ∏ c∈C yc, and e`(ΘC) = em({θc}c∈C) stands for the degree ` elementary symmetric polynomial of the elements {θc}c∈C , ek({θc}c∈C) = δ0,k if k ≤ 0; if B ⊂ [1, n], \|B\| = 2s, we set MB({ψij) = ∏ L⊂B, \|L\|=s (i1,...,is)⊂L (j1,...,js)⊂B\L, iα<jα,jα∈B\L, iα<n ∀α ψiα,jα ; in a similar manner one can define MB({qij}); finally we set Hr(A) = ha2r ( ∑ (a1,...,ar−1)⊂A\{a2r} r−1∏ j=1 max(aj − 2j + 1, 0) haj ) . It is not difficult to show that Hr(A) ∣∣ ha=1, a∈A = (2r − 1)!!, as well as the number of different monomials which appear in Hr([1, 2r]) is equal to the Catalan number Catr. For example, H3([1, 6]) = h6(h24 + 2h25 + 2h34 + 4h35 + 6h45), H4([1, 8]) = h8 ( h246 + 2h247 + 2h256 + 4h257 + 6h267 + 2h346 + 4h347 + 4h356 + 8h357 + 12h367 + 6h456 + 12h457 + 18h467 + 24h567 ) . Exercise 3.9. Write Hr([1, 2r]) = ha2r ( ∑ A:=(a1,...,ar−1)⊂[1,2r−1] aj≥2j c (r) A hA ) , where c (r) A := r−1∏ j=1 max(aj − 2j + 1, 0) and hA := ∏ a∈A ha. Show that ∑ A:=(a1,...,ar−1)⊂[1,2r−1] aj≥2j ( c (r) A )2 = Er, (3.3) where Er denotes the r-th Euler number, see, e.g., [131, A000364]. Find representation theoretic interpretation of numbers {c(r) A } and the identity (3.3). 44 A.N. Kirillov Clearly, ∑ A:=(a1,...,ar−1)⊂[1,2r−1] aj≥2j c (r) A = (2r − 1)!!. Question 3.10. Does there exist a semisimple algebra A(r), dim(A(r)) = Er such that the all irreducible representations π (r) A of the algebra A(r) are in one-to-one correspondence with the set P(r) := {A = (a1, . . . , ar−1) ⊂ [1, 2r − 1], aj ≥ 2j, ∀ j} and dim(πA) = c (r) A , ∀A ∈ P(r)? It is worth noting that the Dunkl element θi, 1 ≤ i ≤ n, doesn’t commute either with yj , j 6= i or any ψkl. On the other hand one can check easily that [ek(y1, . . . , yn), θi] = 0, ∀ k, i. 3.2 Multiplicative Dunkl elements Since the elements uij , uik and ujk, i < j < k, satisfy the classical and quantum Yang–Baxter relations (3.1) and (3.2), one can define a multiplicative analogue denoted by Θi, 1 ≤ i ≤ n, of the Dunkl elements θi. Namely, to start with, we define elements hij := hij(t) = 1 + tuij , i 6= j. We consider hij(t) as an element of the algebra 3̃HTn := 3HTn(β) ⊗ Z[[q±1 ij , t, x, y, . . .]], where we assume that the all parameters {qij , t, x, y, . . .} are central in the algebra 3̃HTn. Lemma 3.11. (1a) hij(x)hij(y) = hij(x+ y + βxy) + qijxy, (1b) hij(x)hji(y) = hij(x− y) + βy − qijxy if i < j. It follows from (1b) that hij(t)hji(t) = 1 + βt − t2qij if i < j, and therefore the elements {hij} are invertible in the algebra 3̃HTn. (2) hij(x)hjk(y) = hjk(y)hik(x) + hik(y)hij(x)− hik(x+ y + βxy), (3) multiplicative Yang–Baxter relations: hijhikhjk = hjkhikhij if i < j < k, (4) define multiplicative Dunkl elements (in the algebra 3̃HTn) as follows Θj := Θj(t) =  1∏ a=j−1 h−1 aj (j+1∏ a=n hja ) , 1 ≤ j ≤ n. Then the multiplicative Dunkl elements pair-wise commute. Clearly n∏ j=1 Θj = 1, Θj = 1 + tθj + t2(· · · ) and ΘI ∏ i/∈I, j∈I i<j ( 1 + tβ − t2qij ) ∈ 3HTn(β). Here for a subset I ⊂ [1, n] we use notation ΘI = ∏ a∈I Θa. Note, that the element ΘI is a product of (exactly!) k(n− k) terms of a form hiαjα , where k := \|I\|. Our main result of this section is a description of relations among the multiplicative Dunkl elements. On Some Quadratic Algebras 45 Theorem 3.12 (A.N. Kirillov and T. Maeno [76]). In the algebra 3HTn(β) the following rela- tions hold true∑ I⊂[1,n] \|I\|=k ΘI ∏ i/∈I, j∈J i<j ( 1 + tβ − t2qij ) = [ n k ] 1+tβ . Here [ n k ] q denotes the q-Gaussian polynomial. Corollary 3.13. Assume that qij 6= 0 for all 1 ≤ i < j ≤ n. Then the all elements {uij} are invertible and u−1 ij = q−1 ij (uij − β). Now define elements Φi ∈ 3̃HTn as follows Φi = { 1∏ a=i−1 u−1 ai }{ i+1∏ a=n uia } , i = 1, . . . , n. Then we have (1) relationship among Θj and Φj: tn−2j+1Θj ( t−1 ) \|t=0 = (−1)jΦj , (2) the elements {Φi, 1 ≤ i ≤ n, } generate a commutative subalgebra in the algebra 3̃HTn, (3) for each k = 1, . . . , n, the following relation in the algebra 3HTn among the elements {Φi} holds ∑ I⊂[1,n] \|I\|=k ∏ i/∈I, j∈I i<j (−qij)ΦI = βk(n−k), where ΦI := ∏ a∈I Φa. In fact the element Φi admits the following “reduced expression” (i.e., one with the minimal number of terms involved) which is useful for proofs and applications Φi = {−→∏ j∈I {−−→∏ i∈Ic+ i<j u−1 ij }}{−−→∏ j∈Ic+ {−→∏ i∈I i<j uij }} . (3.4) Let us explain notations. For any (totally) ordered set I = (i1 < i2 < · · · < ik) we denote by I+ the set I with the opposite order, i.e., I+ = (ik > ik−1 > · · · > i1); if I ⊂ [1, n], then we set Ic := [1, n]\I. For any (totally) ordered set I we denote by −→∏ i∈I the ordered product according to the order of the set I. Note that the total number of terms in the r.h.s. of (3.4) is equal to i(n− i). Finally, from the “reduced expression” (3.4) for the element Φi one can see that ∏ i/∈I,j∈I i<j (−qij)ΦI = {−→∏ j∈I {−−→∏ i∈Ic+ i<j (β − uij) }}{−−→∏ j∈Ic+ {−→∏ i∈I i<j uij }} := Φ̃I ∈ 3HTn. Therefore the identity∑ I⊂[1,n] \|I\|=k Φ̃I = βk(n−k) is true in the algebra 3HTn for any set of parameters {qij}. 46 A.N. Kirillov Comments 3.14. In fact from our proof of Theorem 3.8 we can deduce more general statement, namely, consider integers m and k such that 1 ≤ k ≤ m ≤ n. Then∑ I⊂[1,m] \|I\|=k ΘI ∏ i∈[1,m]\I, j∈J i<j ( 1 + tβ − t2qij ) = [ m k ] 1+tβ + ∑ A⊂[1,n], B⊂[1,n] \|A\|=\|B\|=r uA,B, (3.5) where, by definition, for two sets A = (i1, . . . , ir) and B = (j1, . . . , jr) the symbol uA,B is equal to the (ordered) product r∏ a=1 uia,ja . Moreover, the elements of the sets A and B have to satisfy the following conditions: • for each a = 1, . . . , r one has 1 ≤ ia ≤ m < ja ≤ n, and k ≤ r ≤ k(n− k). Even more, if r = k, then sets A and B have to satisfy the following additional conditions: • B = (j1 ≤ j2 ≤ · · · ≤ jk), and the elements of the set A are pair-wise distinct. In the case β = 0 and r = k, i.e., in the case of additive (truncated) Dunkl elements, the above statement, also known as the quantum Pieri formula, has been stated as conjecture in [45], and has been proved later in [117]. Corollary 3.15 ([76]). In the case when β = 0 and qij = qiδj−i,1, the algebra over Z[q1, . . . , qn−1] generated by the multiplicative Dunkl elements {Θi and Θ−1 i , 1 ≤ i ≤ n} is canonically isomor- phic to the quantum K-theory of the complete flag variety F ln of type An−1. It is still an open problem to describe explicitly the set of monomials {uA,B} which appear in the r.h.s. of (3.5) when r > k. 3.3 Truncated Gaudin operators Let {pij , 1 ≤ i 6= j ≤ n} be a set of mutually commuting parameters. We assume that parame- ters {pij}1≤i<j≤n are invertible and satisfy the Arnold relations 1 pik = 1 pij + 1 pjk , i < j, k. For example one can take pij = (zi − zj)−1, where z = (z1, . . . , zn) ∈ (C\0)n. Definition 3.16. Truncated (rational) Gaudin operator corresponding to the set of parame- ters {pij} is defined to be Gi = ∑ j 6=i p−1 ij sij , 1 ≤ i ≤ n, where sij denotes the exchange operator which switches variables xi and xj , and fixes parame- ters {pij}. We consider the Gaudin operator Gi as an element of the group ring Z[{p±1 ij }][Sn], call this element Gi ∈ Z[{p±1 ij }][Sn], i = 1, . . . , n, by Gaudin element and denoted it by θ (n) i . It is easy to see that the elements uij := p−1 ij sij , 1 ≤ i 6= j ≤ n, define a representation of the algebra 3HTn(β) with parameters β = 0 and qij = u2 ij = p2 ij . Therefore one can consider the (truncated) Gaudin elements as a special case of the (trun- cated) Dunkl elements. Now one can rewrite the relations among the Dunkl elements, as well as the quantum Pieri formula [45, 117], in terms of the Gaudin elements. On Some Quadratic Algebras 47 The key observation which allows to rewrite the quantum Pieri formula as a certain relation among the Gaudin elements, is the following one: parameters { p−1 ij } satisfy the Plücker relations 1 pikpjl = 1 pijpkl + 1 pilpjk if i < j < k < l. To describe relations among the Gaudin elements θ (n) i , i = 1, . . . , n, we need a bit of notation. Let {pij} be a set of invertible parameters as before, ia < ja, a = 1, . . . , r. Define polynomials in the variables h = (h1, . . . , hn) G (n) m,k,r(h, {pij}) = ∑ I⊂[1,n−1] \|I\|=r 1∏ i∈I pin ∑ J⊂[1,n] \|I\|+m=\|J\|+k ( n− \|I ∪ J \| n−m− \|I\| ) h̃J , (3.6) where h̃J = ∑ K⊂J, L⊂J \|K\|=\|L\|, K ⋂ L=∅ ∏ j∈J\(K∪L) hj ∏ ka∈K, la∈L p2 ka,la , and summation runs over subsets K = {k1 < k2 < · · · < kr} and L = {l1 < l2 < · · · < lr} ⊂ J}, such that ka < la, a = 1, . . . , r. Theorem 3.17 (relations among the Gaudin elements [72], cf. [108]). (1) Under the assumption that elements {pij , 1 ≤ i < j ≤ n} are invertible, mutually commute and satisfy the Arnold relations, one has G (n) m,k,r ( θ (n) 1 , . . . , θ(n) n , {pij} ) = 0 if m > k, G (n) 0,0,r ( θ (n) 1 , . . . , θ(n) n , {pij} ) = er(d2, . . . , dn), (3.7) where d2, . . . , dn denote the Jucys–Murphy elements in the group ring Z[Sn] of the sym- metric group Sn, see Comments 3.4 for a definition of the Jucys–Murphy elements. (2) Let J = {j1 < j2 < · · · < jr} ⊂ [1, n], define matrix MJ := (ma,b)1≤a,b≤r, where ma,b := ma,b(h; {pij}) =  hja if a = b, pja,jb if a < b, −pjb,ja if a > b. Then h̃J = DET \|MJ \|. Examples 3.18. (1) Let us display the polynomials G (n) m,k,r(h, {pij}) a few cases G (n) m,0,r(h, {pij}) = ∑ I⊂[1,n−1] \|I\|=r ∏ i∈I p−1 in ( ∑ J⊂[1,n] \|J\|=m+r, I⊂J h̃J ) , G (n) m,k,0(h, {pij}) = ( n−m+ k k ) eqm−k(h1, . . . , hn), G (n) m,1,r(h, {pij}) = ∑ I⊂[1,n−1] \|I\|=r ∏ i∈I p−1 in ( ∑ J⊂[1,n] I⊂J, \|J\|=m+r (n−m− r + 1)h̃J + ∑ J⊂[1,n] \|J\|=m+r−1, \|I∪J\|=m+r h̃J ) . 48 A.N. Kirillov (2) Let us list the relations (3.7) among the Gaudin elements in the case n = 3. First of all, the Gaudin elements satisfy the “standard” relations among the Dunkl elements θ1 +θ2 +θ3 = 0, θ1θ2 + θ1θ3 + θ2θ3 + q12 + q13 + q23 = 0, θ1θ2θ3 + q12θ3 + q13θ2 + q23θ1 = 0. Moreover, we have additional relations which are specific for the Gaudin elements G (3) 2,0,1 = 1 p13 (θ1θ2 + θ1θ3 + q12 + q13) + 1 p23 (θ1θ2 + θ2θ3 + q12 + q23) = 0, the elements p23θ1 + p13θ2 and θ1θ2 are central. It is well-known that the elementary symmetric polynomials er(d2, . . . , dn) := Cr, r = 1, . . . , n − 1, generate the center of the group ring Z[p±1 ij ][Sn], whereas the Gaudin elements {θ(n) i , i = 1, . . . , n}, generate a maximal commutative subalgebra B(pij), the so-called Bethe subalgebra, in Z[p±1 ij ][Sn]. It is well-known, see, e.g., [108], that B(pij) = ⊕ λ`n Bλ(pij), where Bλ(pij) is the λ-isotypic component of B(pij). On each λ-isotypic component the value of the central element Ck is the explicitly known constant ck(λ). It follows from [108] that the relations (3.7) together with relations G0,0,r ( θ (n) 1 , . . . , θ(n) n , {pij} ) = cr(λ) are the defining relations for the algebra Bλ(pij). Let us remark that in the definition of the Gaudin elements we can use any set of mutually commuting, invertible elements {pij} which satisfies the Arnold conditions. For example, we can take pij := qj−2(1− q) 1− qj−i , 1 ≤ i < j ≤ n. It is not difficult to see that in this case lim q→0 θ (n) J p1j = −dj = − j−1∑ a=1 saj , where as before, dj denotes the Jucys–Murphy element in the group ring Z[Sn] of the symmetric group Sn. Basically from relations (3.7) one can deduce the relations among the Jucys–Murphy elements d2, . . . , dn after plugging in (3.6) the values pij := qj−2(1−q) 1−qj−i and passing to the limit q → 0. However the real computations are rather involved. Finally we note that the multiplicative Dunkl/Gaudin elements {Θi, 1, . . . , n} also generate a maximal commutative subalgebra in the group ring Z[p±1 ij ][Sn]. Some relations among the elements {Θl} follow from Theorem 3.12, but we don’t know an analogue of relations (3.7) for the multiplicative Gaudin elements, but see [108]. Exercises 3.19. Let A = (ai,j) be a 2m×2m skew-symmetric matrix. The Pfaffian and Hafnian of A are defined correspondingly by the equations Pf(A) = 1 2mm! ∑ σ∈S2m sgn(σ) m∏ i=1 aσ(2i−1),σ(2i), Hf(A) = 1 2mm! ∑ σ∈S2m m∏ i=1 aσ(2i−1),σ(2i), where S2m is the symmetric group and sgn(σ) is the signature of a permutation σ ∈ S2m.30 30See, e.g., https://en.wikipedia.org/wiki/Pfaffian. https://en.wikipedia.org/wiki/Pfaffian On Some Quadratic Algebras 49 Now let n be a positive integer, and {pij , 1 ≤ i 6= j ≤ n, pij + pji = 0} be a set of skew- symmetric, invertible and mutually commuting elements. We set pii = 0 for all i, and q :={ p2 ij } 1≤i<j≤n. Now let us assume that the elements {pij}1≤i<j≤n satisfy the Plüker relations for the elements{ p−1 ij } 1≤i<j≤n, namely, 1 pikpjl = 1 pijpkl + 1 pilpjk for all 1 ≤ i < j < k < l ≤ n. (a) Let n be an even positive integer. Let us define An(pij) := (pij)1≤i,j≤n to be the n × n skew-symmetric matrix corresponding to the family {pij}1≤i<j≤n. Show that DET \|An(pij)\| = Hf ( An ( p2 ij )) . (b) Let n be a positive integer, and z1, . . . , zn be a set of mutually commuting variables, define polynomials Hi(z1, . . . , zn \| {pij}), i = 1, . . . , n from the equation DET \| diag(t+ z1, . . . , t+ zn) +An(pij)\| = tn + n∑ i=1 Hi(z1, . . . , zn \| {pij})tn−i, where diag(t+ z1, . . . , t+ zn) means the diagonal matrix. Show that for k = 1, . . . , n the polynomialHk(z1, . . . , zn \| {pij}) is equal to the multiparameter quantum elementary polynomial e (q) k (z1, . . . , zn), see, e.g., [45], or Theorem 2.7. For example, take n = 4, then DET \|A(pij)\| = (p12p34 − p13p24 + p14p23)2 = p2 12p 2 34 + p2 13p 2 24 + p2 14p 2 23 − 2p12p13p23p14p24p34 ( 1 p12p34 − 1 p13p24 + 1 p14p23 ) = p2 12p 2 34 + p2 13p 2 24 + p2 14p 2 23 = Hf ( A4 ({ p2 ij })) . The last equality follows from the Plücker relations for parameters {p−1 ij }. On the other hand, if one assumes that a set of skew symmetric parameters {rij}1≤i<j≤n, rij + rji = 0, satisfies the “standard” Plücker relations, namely rikrjl = rijrkl + rilrjk, i < j < k < l, then DET \|An(rij)\| = 0. 3.4 Shifted Dunkl elements di and Di As it was stated in Corollary 3.15, the truncated additive and multiplicative Dunkl elements in the algebra 3HTn(0) generate over the ring of polynomials Z[q1, . . . , qn−1] correspondingly the quantum cohomology and quantum K-theory rings of the full flag variety F ln. In order to describe the corresponding equivariant theories, we will introduce the shifted additive and multiplicative Dunkl elements. To start with we need at first to introduce an extension of the algebra 3HTn(β). Let {z1, . . . , zn} be a set of mutually commuting elements and {β,h = (h2, . . . , hn), t, qij = qji, 1 ≤ i, j ≤ n} be a set of parameters. We set hn := 0. Definition 3.20 (cf. Definition 3.1). Define algebra 3THn(β,h) to be the semi-direct product of the algebra 3THn(β) and the ring of polynomials Z[h, t][z1, . . . , zn] with respect to the crossing relations 50 A.N. Kirillov (1) ziukl = uklzi if i /∈ {k, l}, (2) ziuij = uijzj + βzi + hj , zjuij = uijzi − βzi − hj if 1 ≤ i < j < k ≤ n. Now we set as before hij := hij(t) = 1 + tuij . Definition 3.21. • Define shifted additive Dunkl elements to be di = zi + ∑ i<j uij − ∑ i>j uji. • Define shifted multiplicative Dunkl elements to be Di = ( 1∏ a=i−1 h−1 ai ) (1 + zi) ( i+1∏ a=n hia ) . Lemma 3.22. [di, dj ] = 0, [Di,Dj ] = 0 for all i, j. Now we stated an analogue of Theorem 3.8 for shifted multiplicative Dunkl elements. As a preliminary step, for any subset I ⊂ [1, n] let us set DI = ∏ a∈I Da. It is clear that DI ∏ i/∈I, j∈I i<j ( 1 + tβ − t2qij ) ∈ 3HTn(β,h). Theorem 3.23. In the algebra 3HTn(β,h) the following relations hold true∑ I⊂[1,n] \|I\|=k DI ∏ i/∈I,j∈J i<j ( 1 + tβ − t2qij ) = ∑ I⊂[1,n] I={1≤i1<...<ik≤n} k∏ a=1 (1 + tβ)n−k−ia+a ( zia(1 + tβ)ia−a + 1 + hia (1 + tβ)ia−a − 1 β ) . In particular, if β = 0, we will have Corollary 3.24. In the algebra 3HTn(0,h) the following relations hold ∑ I⊂[1,n] \|I\|=k DI ∏ i/∈I,j∈J i<j ( 1− t2qij ) = ∑ I⊂[1,n] I={1≤i1,...,ik≤n} k∏ a=1 ( zia + 1 + thia(ia − a) ) . (3.8) Conjecture 3.25. If h1 = · · · = hn−1 = 1, t = 1 and qij = δi,j+1, then the subalgebra generated by multiplicative Dunkl elements Di, i = 1, . . . , n, in the algebra 3HTn(0,h = 1) (and t = 1), is isomorphic to the equivariant quantum K-theory of the complete flag variety F ln. Our proof is based on induction on k and the following relations in the algebra 3HTn(β,h) hji · (1 + xj) = hj + (1 + β)xj − xi + (1 + xi)hji, hjihjk = hjkhki + hikhji − 1− β, On Some Quadratic Algebras 51 if i < j < k, and we set hij := hij(1). These relations allow to reduce the left hand side of the relations listed in Theorem 3.23 to the case when zi = 0, hi = 0, ∀ i. Under these assumptions one needs to proof the following relations in the algebra 3HTn(β), see Theorem 3.12, ∑ I⊂[1,n] \|I\|=k DI ∏ i/∈I, j∈J i<j ( 1 + tβ − t2qij ) = [ n k ] 1+tβ . In the case β = 0 the identity (3.8) has been proved in [76]. One of the main steps in our proof of Theorem 3.8 is the following explicit formula for the elements DI . Lemma 3.26. One has D̃I := DI ∏ i/∈I, j∈I i<j ( 1 + tβ − t2qij ) = ↗∏ b∈I ( ↘∏ a/∈I a<b hba ) ↗∏ a∈I ( (1 + za) ↘∏ b/∈I a<b hab ) . Note that if a < b, then hba = 1 + βt− uab. Here we have used the symbol ↗∏ b∈I ( ↘∏ a/∈I a<b hba ) to denote the following product. At first, for a given element b ∈ I let us define the set I(b) := {a ∈ [1, n]\I, a < b} := (a (b) 1 < · · · < a (b) p ) for some p (depending on b). If I = (b1 < b2 < · · · < bk), i.e., bi = a (b) i , then we set ↗∏ b∈I ( ↘∏ a/∈I a<b hba ) = k∏ j=1 ( ubj ,asubj ,as−1 · · ·ubj ,a1 ) . For example, let us take n = 6 and I = (1, 3, 5), then D̃I = h32h54h52(1 + z1)h16h14h12(1 + z3)h36h34(1 + z5)h56. Let us stress that the element D̃I ∈ 3HTn(β) is a linear combination of square free monomials and therefore, a computation of the left hand side of the equality stated in Theorem 3.17 can be performed in the “classical case” that is in the case qij = 0, ∀ i < j. This case corresponds to the computation of the classical equivariant cohomology of the type An−1 complete flag variety F ln if h = 1. A proof of the β = 0 case given in [76, Theorem 1], can be immediately extended to the case β 6= 0. Exercises 3.27. (1) Show that ∑ 1≤i1<···<ik≤n k∏ a=1 (1 + β)n−k−ia+a = [ n k ] 1+tβ . 52 A.N. Kirillov (2) (β, h)-Stirling polynomials of the second type. Define polynomials Sn,k(β, h) as follows Sn,k(β, h) = ∑ I⊂[1,n] I={1≤i1,...,ik≤n} k∏ a=1 ( βn−k−ia+a + h βn−k−ia+a − 1 β − 1 ) . Show that Sn,k(1, 1) = { n+ 1 k + 1 } , Sn,k(β, 0) = [ n k ] β . 4 Algebra 3T (0) n (Γ) and Tutte polynomial of graphs 4.1 Graph and nil-graph subalgebras, and partial flag varieties Let’s consider the set Rn := {(i, j) ∈ Z× Z \| 1 ≤ i < j ≤ n} as the set of edges of the complete graph Kn on n labeled vertices v1, . . . , vn. Any subset S ⊂ Rn is the set of edges of a unique subgraph Γ := ΓS of the complete graph Kn. Definition 4.1 (graph and nil-graph subalgebras). The graph subalgebra 3Tn(Γ) (resp. nil- graph subalgebra 3T (0) n (Γ)) corresponding to a subgraph Γ ⊂ Kn of the complete graph Kn, is defined to be the subalgebra in the algebra 3Tn (resp. 3T (0) n ) generated by the elements {uij \| (i, j) ∈ Γ}. In subsequent Sections 4.1.1 and 4.1.2 we will study some examples of graph subalgebras corresponding to the complete multipartite graphs, cycle graphs and linear graphs. 4.1.1 Nil-Coxeter and affine nil-Coxeter subalgebras in 3T (0) n Our first example is concerned with the case when the graph Γ corresponds to either the set S := {(i, i+ 1) \| i = 1, . . . , n− 1} of simple roots of type An−1, or the set Saff := S ∪ {(1, n)} of affine simple roots of type A (1) n−1. Definition 4.2. (a) Denote by ÑCn subalgebra in the algebra 3T (0) n generated by the elements ui,i+1, 1 ≤ i ≤ n− 1. (b) Denote by ÃNCn subalgebra in the algebra 3T (0) n generated by the elements ui,i+1, 1 ≤ i ≤ n− 1 and −u1,n. Theorem 4.3. (A) The subalgebra ÑCn is canonically isomorphic to the nil-Coxeter algebra NCn. In partic- ular, Hilb(ÑCn, t) = [n]t! (cf. [8]). (B) The subalgebra ÃNCn has finite dimension and its Hilbert polynomial is equal to Hilb(ÃNCn, t) = [n]t ∏ 1≤j≤n−1 [j(n− j)]t = [n]t! ∏ 1≤j≤n−1 [j]tn−j . In particular, dim ÃNCn = (n− 1)!n!, degt Hilb(ÃNCn, t) = ( n+1 3 ) . On Some Quadratic Algebras 53 (C) The kernel of the map π : ÃNCn −→ ÑCn, π(u1,n) = 0, π(ui,i+1) = ui,i+1, 1 ≤ i ≤ n− 1, is generated by the following elements: f (k) n = 1∏ j=k n−k+j−1∏ a=j ua,a+1, 1 ≤ k ≤ n− 1. Note that deg f (k) n = k(n− k). The statement (C) of Theorem 4.3 means that the element f (k) n which does not contain the generator u1,n, can be written as a linear combination of degree k(n − k) monomials in the algebra ÃNCn, each contains the generator u1,n at least once. By this means we obtain a set of all extra relations (i.e., additional to those in the algebra ÑCn) in the algebra ÃNCn. More- over, each monomial M in all linear combinations mentioned above, appears with coefficient (−1)#\|u1,n∈M \|+1. For example, f (1) 4 := u1,2u2,3u3,4 = u2,3u3,4u1,4 + u3,4u1,4u1,2 + u1,4u1,2u2,3, f (2) 4 := u2,3u3,4u1,2u2,3 = u1,2u3,4u2,3u1,4 + u1,2u2,3u1,4u1,2 + u2,3u1,4u1,2u3,4 + u3,4u2,3u1,4u3,4 − u1,4u1,2u3,4u1,4. Worthy of mention is that dim(ÃNCn) = (n− 1)!n! is equal to the number of (directed) Hamil- tonian cycles in the complete bipartite graph Kn,n, see, e.g., [131, A010790] for additional information. Remark 4.4. More generally, let (W,S) be a finite crystallographic Coxeter group of rank l with the set of exponents 1 = m1 ≤ m2 ≤ · · · ≤ ml. Let BW be the corresponding Nichols–Woronowicz algebra, see, e.g., [8]. Follow [8], denote by ÑCW the subalgebra in BW generated by the elements [αs] ∈ BW corresponding to simple roots s ∈ S. Denote by ÃNWCW the subalgebra in BW generated by ÑCW and the element [aθ], where [aθ] stands for the element in BW corresponding to the highest root θ for W . In other words, ÃNWCW is the image of the algebra ÃNCW under the natural map BE(W ) −→ BW , see, e.g., [8, 73]. It follows from [8, Section 6], that Hilb(ÑCW , t) = l∏ i=1 [mi + 1]t. Conjecture 4.5 (Yu. Bazlov and A.N. Kirillov, 2002). Hilb ( ÃNWCW , t ) = l∏ i=1 1− tmi+1 1− tmi l∏ i=1 1− tai 1− t = Paff(W, t) l∏ i=1 (1− tai), where Paff(W, t) := ∑ w∈Waff tl(w) = l∏ i=1 (1 + t+ · · ·+ tmi) 1− tmi denotes the Poincaré polynomial corresponding to the affine Weyl group Waff , see [17, p. 245]; ai := (2ρ, α∨i ), 1 ≤ i ≤ l, denote the coefficients of the decomposition of the sum of positive roots 2ρ in terms of the simple roots αi. In particular, dim ÃNWCW = \|W \| l∏ i=1 ai l∏ i=1 mi and deg Hilb ( ÃNWCW , t ) = l∑ 1=1 ai. 54 A.N. Kirillov It is well-known that the product l∏ i=1 1−tai 1−tmi is a symmetric (and unimodal?) polynomial with non-negative integer coefficients. Example 4.6. (a) Hilb ( ÃNC3, t ) = [2]2t [3]t, Hilb ( ÃNC4, t ) = [3]2t [4]2t , Hilb ( ÃNC5, t ) = [4]2t [5]t[6]2t , (b) Hilb(BE2, t) = (1 + t)4 ( 1 + t2 )2 , Hilb(ÃNCB2 , t) = (1 + t)3 ( 1 + t2 )2 = Paff(B2, t) ( 1− t3 )( 1− t4 ) . (c) Hilb ( ÃNCB3 , t ) = (1 + t)3 ( 1 + t2 )2( 1 + t3 )( 1 + t4 )( 1 + t+ t2 )( 1 + t3 + t6 ) = Paff(B3, t) ( 1− t5 )( 1− t8 )( 1− t9 ) . Indeed, mB3 = (1, 3, 5), aB3 = (5, 8, 9). Definition 4.7. Let 〈ÃNCn〉 denote the two-sided ideal in 3T (0) n generated by the elements {ui,i+1}, 1 ≤ i ≤ n− 1, and u1,n. Denote by Un the quotient Un = 3T 0 n/〈ÃNCn〉. Proposition 4.8. U4 ∼= 〈u1,3, u2,4〉 ∼= Z2 × Z2, U5 ∼= 〈u1,4, u2,4, u2,5, u3,5, u1,3〉 ∼= ÃNC5. In particular, Hilb ( 3T (0) 5 , t ) = [ Hilb(ÃNC5, t) ]2 . 4.1.2 Parabolic 3-term relations algebras and partial f lag varieties In fact one can construct an analogue of the algebra 3HTn and a commutative subalgebra inside it, for any graph Γ = (V,E) on n vertices, possibly with loops and multiple edges [72]. We denote this algebra by 3Tn(Γ), and denote by 3T (0) n (Γ) its nil-quotient, which may be considered as a “classical limit of the algebra 3Tn(Γ)”. The case of the complete graph Γ = Kn reproduces the results of the present paper and those of [72], i.e., the case of the full flag variety F ln. The case of the complete multipartite graph Γ = Kn1,...,nr reproduces the analogue of results stated in the present paper for the full flag variety F ln, to the case of the partial flag variety Fn1,...,nr , see [72] for details. We expect that in the case of the complete graph with all edges having the same multiplic- ity m, denoted by either Γ = K (m) n , or mKn in the present paper, the commutative subalgebra generated by the Dunkl elements in the algebra 3T (0) n (Γ) is related to the algebra of coinvariants of the diagonal action of the symmetric group Sn on the ring of polynomials Q [ X (1) n , . . . , X (m) n ] , where we set X (i) n = { x (i) 1 , . . . , x (i) n } . Example 4.9. Take Γ = K2,2. The algebra 3T (0)(Γ) is generated by four elements {a = u13, b = u14, c = u23, d = u24} subject to the following set of (defining) relations • a2 = b2 = c2 = d2 = 0, cb = bc, ad = da, • aba+ bab = 0 = aca+ cac, bdb+ dbd = 0 = cdc+ dcd, abd− bdc− cab+ dca = 0 = acd− bac− cdb+ dba, • abca+ adbc+ badb+ bcad+ cadc+ dbcd = 0. On Some Quadratic Algebras 55 It is not difficult to see that31 Hilb ( 3T (0)(K2,2), t ) = [3]2t [4]2t , Hilb ( 3T (0)(K2,2)ab, t ) = (1, 4, 6, 3). Here for any algebra A we denote by Aab its abelianization32. The commutative subalgebra in 3T (0)(K2,2), which corresponds to the intersection 3T (0)(K2,2) ∩Z[θ1, θ2, θ3, θ4], is generated by the elements c1 := θ1 + θ2 = (a + b + c + d) and c2 := θ1θ2 = (ac+ ca+ bd+db+ad+ bc). The elements c1 and c2 commute and satisfy the following relations c3 1 − 2c1c2 = 0, c2 2 − c2 1c2 = 0. The ring of polynomials Z[c1, c2] is isomorphic to the cohomology ring H∗(Gr(2, 4),Z) of the Grassmannian variety Gr(2, 4). To continue exposition, let us take m ≤ n, and consider the complete multipartite graph Kn,m which corresponds to the Grassmannian variety Gr(n,m+ n). One can show Hilb ( 3T (0) n+m(Kn,m)ab, t ) = n−1∑ k=0 (−1)k(1 + (n− k)t)m−1 n−k∏ j=1 (1 + jt) { n n− k } = tn+m−1Tutte ( Kn,m, 1 + t−1, 0 ) , where { n k } := S(n, k) denotes the Stirling numbers of the second kind, that is the number of ways to partition a set of n labeled objects into k nonempty unlabeled subsets, and for any graph Γ, Tutte(Γ, x, y) denotes the Tutte polynomial33 corresponding to graph Γ. It is well-known that the Stirling numbers S(n, k) satisfy the following identities n−1∑ k=0 (−1)k S(n, n− k) n−k∏ j=1 (1 + jt) = (1 + t)n, ∑ n≥k { n k } xn n! = (ex − 1)k k! . Let us observe that dim ( 3T (0)(Kn,n)ab ) = n−1∑ k=0 (−1)k(n+ 1− k)n−1(n+ 1− k)! { n n− k } = n+1∑ k=1 ((k − 1)!)2 { n+ 1 k }2 , see, e.g., [131, A048163]. Moreover, if m ≥ 0, then∑ n≥1 dim ( 3T (0)(Kn,n+m)ab ) tn = ∑ k≥1 kk+m−1(k − 1)!tk k−1∏ j=1 (1 + kjt) , ∑ n≥1 Hilb ( 3T (0)(Kn,m)ab, t ) zn−1 = ∑ k≥0 (1 + kt)m−1 k∏ j=1 z(1 + jt) 1 + jz . 31Hereinafter we shell use notation (a0, a1, . . . , ak)t := a0 + a1t+ · · ·+ akt k. 32See http://groupprops.subwiki.org/wiki/Abelianization. 33See, e.g., https://en.wikipedia.org/wiki/Tutte_polynomial. It is well-known that Tutte(Γ, 1 + t, 0) = (−1)\|Γ\|t−κ(Γ)Chrom(Γ,−t), where for any graph Γ, \|Γ\| is equal to the number of vertices and κ(Γ) is equal to the number of connected components of Γ. Finally Chrom(Γ, t) denotes the chromatic polynomial corresponding to graph Γ, see, e.g., [140], or https://en.wikipedia.org/wiki/Chromatic_polynomial. http://groupprops.subwiki.org/wiki/Abelianization https://en.wikipedia.org/wiki/Tutte_polynomial https://en.wikipedia.org/wiki/Chromatic_polynomial 56 A.N. Kirillov Comments 4.10 (poly-Bernoulli numbers). Based on listed above identities involving the Stir- ling numbers S(n, k), one can prove the following combinatorial formula dim ( 3T (0)(Kn,m)ab ) = min(n,m)∑ j=0 (j!)2 { n+ 1 j + 1 }{ m+ 1 j + 1 } = B(−m) n = B(−n) m , (4.1) where B (k) n denotes the poly-Bernoulli number introduced by M. Kaneko [64]. On the other hand, it is well-known, see, e.g., footnote 33, that for any graph Γ the spe- cialization Tutte(Γ; 2, 0) of the Tutte polynomial associated with graph Γ, counts the number of acyclic orientations of Γ. Therefore, the poly-Bernulli number B (−m) n counts the number of acyclic orientatations of the complete bipartite graph Kn,m. For example, dim ( 3T (0)(K3,3)ab ) = 230 = 1 + 72 + (2!)262 + (3!)2, cf. Example 4.16(3). For the reader’s convenient, we recall below a definition of poly-Bernoulli numbers. To start with, let k be an integer, consider the formal power series Lik(z) := ∞∑ n=1 zn nk . If k ≥ 1, Lik(z) is the k-th polylogarithm, and if k ≤ 0, then Lik(z) is a rational function. Clearly Li1(z) = − ln(1− z). Now define poly-Bernoulli numbers through the generating function Lik(1− e−z) 1− e−z = ∞∑ n=0 B(k) n zn n! . Note that a combinatorial formula for the numbers B (−k) n stated in (4.1) is a consequence of the following identity [64] ∞∑ n=0 ∞∑ k=0 B(−k) n xn n! zk k! = ex+z 1− (1− ex)(1− ez) . Note that the poly-Bernoulli numbers B (−m) n (= B (−n) m ) have the following combinatorial inter- pretation34, namely, the number B (−m) n , and therefore the dimension of the algebra 3T (0)(Kn,m) is equal to B(−m) n = T (n− 1,m) + T (n,m− 1), where [26] T (n,m) := min(n,m)∑ j=0 j!(j + 1)! { n+ 1 j + 1 }{ m+ 1 j + 1 } is equal to the number of permutations w ∈ Sn+m having the excedance set {1, 2, . . . ,m}. Exercise 4.11. Show that polynomial Hilb(3T (0)(Kn,m), t) has degree n+m− 1, and Coefftn+m−1 ( Hilb ( 3T (0)(Kn,m), t )) = T (n− 1,m− 1). Problem 4.12. To find a bijective proof of the identity (4.1). 34See for example, [131, A136126], [131, A099594] or [26, Theorem 3.1], and the literature quoted therein. Recall, that the excedance set of a permutation π ∈ Sn is the set of indices i, 1 ≤ i ≤ n, such that π(i) > i. On Some Quadratic Algebras 57 Final remark, the explicit expression for the chromatic polynomial of the complete tripartite graph Kn,n,n can be found in [131, A212220]. Now let θ (n+m) i = ∑ j 6=i uij , 1 ≤ i ≤ n+m, be the Dunkl elements in the algebra 3T (0)(Kn+m), define the following elements the in the algebra 3T (0)(Kn,m) ck := ek ( θ (n+m) 1 , . . . , θ(n+m) n ) , 1 ≤ k ≤ n, cr := er ( θ (n+m) n+1 , . . . , θ (n+m) n+m ) , 1 ≤ r ≤ m. Clearly,( 1 + n∑ k=1 ckt k )( 1 + m∑ r=1 crt r ) = n+m∏ j=1 ( 1 + θ (n+m) j ) = 1. Moreover, there exist the natural isomorphisms of algebras H∗(Gr(n, n+m),Z) ∼= Z[c1, . . . , cn] /〈( 1 + n∑ k=1 ckt k )( 1 + m∑ r=1 crt r ) − 1 〉 , QH∗(Gr(n, n+m)) ∼= Z[q][c1, . . . , cn] /〈( 1 + n∑ k=1 ckt k )( 1 + m∑ r=1 crt r ) − 1− qtn+m 〉 . Let us recall, see Section 2, footnote 26, that for a commutative ring R and a polynomial p(t) = s∑ j=1 gjt j ∈ R[t], we denote by 〈p(t)〉 the ideal in the ring R generated by the coefficients g1, . . . , gs. These examples are illustrative of the similar results valid for the general complete multipartite graphs Kn1,...,nr , i.e., for the partial flag varieties [72]. To state our results for partial flag varieties we need a bit of notation. Let N := n1 + · · ·+nr, nj > 0, ∀ j, be a composition of sizeN . We setNj := n1+· · ·+nj , j = 1, . . . , r, andN0 = 0. Now, consider the commutative subalgebra in the algebra 3T (0) N (KN ) generated by the set of Dunkl elements { θ (N) 1 , . . . , θ (N) N } , and define elements { c (j,N) kj ∈ 3T (0) N (Kn1,...,nr) } to be the degree kj elementary symmetric polynomials of the Dunkl elements θ (N) Nj−1+1, . . . , θ (N) Nj , namely, c (j) k := c (j,N) kj = ek ( θ (N) Nj−1+1, . . . , θ (N) Nj ) , 1 ≤ kj ≤ nj , j = 1, . . . , r, c (j) 0 = 1, ∀ j. Clearly r∏ j=1 ( nj∑ a=0 c(j) a ta ) = N∏ j=1 ( 1 + θ (N) j tj ) = 1. Theorem 4.13. The commutative subalgebra generated by the elements {c(j) kj , 1 ≤ kj ≤ nj , 1 ≤ j ≤ r−1}, in the algebra 3T (0) N (Kn1,...,nr) is isomorphic to the cohomology ring H∗(F ln1,...,nr ,Z) of the partial flag variety F ln1,...,nr . In other words, we treat the Dunkl elements { θ (N) Nj−1+a, 1 ≤ a ≤ nj } , j = 1, . . . , r, as the Chern roots of the vector bundles {ξj := Fj/Fj−1}, j = 1, . . . , r, over the partial flag variety F ln1,...,nr . 58 A.N. Kirillov Recall that a point F of the partial flag variety F ln1,...,nr , n1 + · · · + nr = N , is a sequence of embedded subspaces F = { 0 = F0 ⊂ F1 ⊂ F2 ⊂ · · · ⊂ Fr = CN } such that dim(Fi/Fi−1) = ni, i = 1, . . . , r. By definition, the fiber of the vector bundle ξi over a point F ∈ F ln1,...,nr is the ni-dimensional vector space Fi/Fi−1. To conclude, let us describe the set of (defining) relations among the elements { c (j) a } , 1 ≤ a ≤ nj , 1 ≤ j ≤ r − 1. To proceed, let us introduce the set of variables { x (j) a \| 1 ≤ a ≤ nj , 1 ≤ j ≤ r − 1 } , and define polynomials b0 = 1, bk := bk ({ x (j) a }) , k ≥ 1 by the use of generating function 1 r−1∏ j=1 nj∏ a=1 ( 1 + x (j) a ) ta = ∑ k≥0 bkt k. Now let us introduce matrix Mm ({ x (j) a }) := (mij), where mij :=  bi+j−1 if j > i, 1 if j = i− 1, i ≥ 2, 0 if j < i− 1. Claim 4.14. detMm ({ c (i) a }) = 0, Nr−1 < m ≤ N . Moreover, H∗(F ln1,...,nr ,Z) ∼= Z[{xja}]/〈MNr−1+1, . . . ,MN 〉. A meaning of the algebra 3T (0) n (Γ) and the corresponding commutative subalgebra inside it for a general graph Γ is still unclear. Conjecture 4.15. 35 (1) Let Γ = (V,E) be a connected subgraph of the complete graph Kn on n vertices. Then Hilb ( 3T (0) n (Γ)ab, t ) = t\|V \|−1Tutte ( Γ; 1 + t−1, 0 ) . (2) Let Γ = (V,E, {mij , (ij) ∈ E}) be a connected subgraph of the complete graph K (m) n with multiple edges such that an edge (ij) ∈ Kn has the multiplicity mij. Let 3T (0) n (Γ,m) de- notes the subalgebra in the algebra 3T (0) n (m) generated by elements {u(α(ij)) ij , (ij) ∈ E, 1 ≤ α(ij) ≤ mij}, see Section 2.2. Let A(Γ, {mij}) denotes the graphic arrangement correspon- ding to the graph (Γ, {mij}), that is the set of hyperplanes {H(ij),a = (xi − xj = a), 0 ≤ a ≤ mij − 1, (ij) ∈ E}. Then 3T (0) n (Γ,m)ab = OS+(A(Γ, {mij})), where for any arrangements of hyperplanes A, OS+(A) denotes the even Orlik–Solomon algebra of the arrangement A [113]. In the case when mij = 1, ∀ 1 ≤ i < j ≤ n, 3T (0) n (Γ)anti = OS(A(Γ)). 35Part (1) of this conjecture has been proved in [89]. In [89] the author has used notation OT(Γ) for the Orlik– Terao algebra associated with (simple) graph Γ. In our paper we prefer to denote the corresponding Orlik–Terao algebra by OS+(Γ). On Some Quadratic Algebras 59 Examples 4.16. (1) Let G = K2,2 be complete bipartite graph of type (2, 2). Then Hilb ( 3T 0 4 (2, 2)ab, t ) = (1, 4, 6, 3) = t2(1 + t) + t(1 + t)2 + (1 + t)3, and the Tutte polynomial for the graph K2,2 is equal to x+ x2 + x3 + y. (2) Let G = K3,2 be complete bipartite graph of type (3, 2). Then Hilb ( 3T 0 5 (3, 2)ab, t ) = (1, 6, 15, 17, 7) = t3(1 + t) + 3t2 (1 + t)2 + 2t(1 + t)3 + (1 + t)4, and the Tutte polynomial for the graph K3,2 is equal to x+ 3x2 + 2x3 + x4 + y + 3xy + y2. (3) Let G = K3,3 be complete bipartite graph of type (3, 3). Then Hilb ( 3T 0 6 (3, 3)ab, t ) = (1, 9, 36, 75, 78, 31) = (1 + t)5 + 4t(1 + t)4 + 10t2(1 + t)3 + 11t3(1 + t)2 + 5t4(1 + t), and the Tutte polynomial of the bipartite graph K3,3 is equal to 5x+ 11x2 + 10x3 + 4x4 + x5 + 15xy + 9x2y + 6xy2 + 5y + 9y2 + 5y3 + y4. (4) Consider complete multipartite graph K2,2,2. One can show that Hilb ( 3T (0) 6 (K2,2,2)ab, t ) = (1, 12, 58, 137, 154, 64) = 11t4(1 + t) + 25t3(1 + t)2 + 20t2(1 + t)3 + 7t(1 + t)4 + (1 + t)5, and Tutte(K2,2,2, x, y) = x(11, 25, 20, 7, 1)x + y(11, 46, 39, 8)x + y2(32, 52, 12)x + y3(40, 24)x + y4(29, 6)x + 15y5 + 5y6 + y7. The above examples show that the Hilbert polynomial Hilb(3T 0 n(G)ab, t) appears to be a cer- tain specialization of the Tutte polynomial of the corresponding graph G. Instead of using the Hilbert polynomial of the algebra 3T 0 n(G)ab one can consider the graded Betti numbers (over a field k) polynomial Bettik(3T 0 n(G)ab, x, y). For example, BettiQ ( 3T 0 3 (K3)ab, x, y ) = 1 + xy(4, 2)x + x2y2(3, 2)x, BettiQ ( 3T 0 4 (K2,2)ab, x, y ) = 1 + 4xy + xy2(1, 9, 3)x + x2y3(1, 6, 3)x, BettiQ ( 3T 0 5 (K3,2)ab, x, y ) = 1 + 6xy + xy2(3, 25, 9) + x2y3(6, 45, 34, 7) + x3y4(3, 25, 25, 7), BettiQ ( 3T 0 4 (K4)ab, x, y ) = 1 + xy(10, 10) + x2y2(25, 46, 26, 6) + x3y3(15, 36, 25, 6), BettiZ/2Z ( 3T 0 4 (K4)ab, x, y ) = 1 + xy(10, 10,1)x + x2y2(25, 46, 26, 6) + x3y3(16, 36, 25, 6), BettiQ ( 3T 0 5 (K5)ab, x, y ) = 1 + xy(20, 30) + x2y2(109, 342, 315, 72) + x3y3(195, 852, 1470, 1232, 639, 190, 24) + x4y4(105, 540, 1155, 1160, 639, 190, 24), BettiQ ( 3T 0 5 (K5)ab, 1, 1 ) = 9304, BettiZ/3Z ( 3T 0 5 (K5)ab, x, y ) = 1 + xy(20, 30) + x2y2(109, 342, 315, 72,1) 60 A.N. Kirillov + x3y3(195, 852, 1471, 1232, 640, 190, 24) + x4y4(105, 540,1156, 1160, 639, 190, 24), BettiZ/3Z ( 3T 0 5 (K5)ab, 1, 1 ) = 9308, BettiZ/2Z ( 3T 0 5 (K5)ab, x, y ) = 1 + xy(20, 30,5) + x2y2(114, 342,340,131,10) + x3y3(220,911,1500, 1291, 649, 190, 24) + x4y4(125,599,1165, 1160, 639, 190, 24), BettiZ/2Z ( 3T 0 5 (K5)ab, 1, 1 ) = 9680, BettiZ/2Z ( 3T 0 6 (K3,3)ab, x, y ) = 1 + 9xy + xy2(9, 69, 27) + x2y3(40, 285, 257,52) + x3y4(59, 526, 866, 563, 201, 31) + x4y5(28,311, 636, 520, 201, 31), BettiZ/2Z ( 3T 0 6 (K3,3)ab, 1, 1 ) = 4740, BettiQ ( 3T 0 6 (K3,3)ab, x, y ) = 1 + 9xy + xy2(9, 69, 27) + x2y3(40, 285, 257, 43) + x3y4(59, 526, 866, 563, 201, 31) + x4y5(28, 302, 636, 520, 201, 31), BettiQ ( 3T 0 6 (K3,3)ab, 1, 1 ) = 4704. Let us observe that in all examples displayed above, the Betti polynomials are divisible by 1+xy. It should be emphasize that in the literatute one can find definitions of big variety of (graded) Betti’s numbers associated with a given simple graph Γ, depending on choosing an algebra/ideal has been attached to graph Γ. For example, to define Betti’s numbers, one can start with edge graph ideal/algebra associated with a graph in question, the Stanley–Reisner ideal/ring and so on and so far. We refer the reader to carefully written book by E. Miller and B. Sturmfels [105] for definitions and results concerning combinatorial commutative algebra graded Betti’s numbers. As far as I’m aware, the graded Betti numbers we are looking for in the present paper, are different from those treated in [105], and more close to those studied in [11]. It is not difficult to see (A.K.) that for a simple connected graph Γ the coefficient just before the (unique!) monomial of the maximal degree in Bettik ( 3T 0(Γ)ab, x, y ) is equal to Tutte(Γ; 1, 0). It is known [10] that the number Tutte(Γ; 1, 0) counts that of acyclic orientations of the edges of Γ with a unique source at a vertex v ∈ Γ, or equivalently [10], the number of maximum Γ-parking functions relative to vertex v. Claim 4.17. Let G = (V,E) be a connected graph without loops. Then over any field k Bettik ( 3T 0 n(G)ab,−x, x ) = (1− x)eHilb ( 3T 0 n(G)ab, x ) , where n = \|V (G)\| = number of vertices, e = \|E(G)\| = number of edges. Question 4.18. • Let G be a connected subgraph of the complete graph Kn. Does the graded Betti polynomial BettiQ(3T 0 n(G)ab, x, y) is a certain specialization of the Tutte polynomial T (G, x, y)? If not, give example of two (simple) graphs such that their Orlik–Terao algebras have the same Tutte polynomial, but different Betti polynomials over Q , and vice versa. • It is clear that for any graph Γ (or matroid) one has Tutte(Γ, x, y) = a(Γ)(x + y) + (higher degree terms) for some integer a(Γ) ∈ N. Does the number a(Γ) have a simple combinatorial interpretation? On Some Quadratic Algebras 61 Proposition 4.19. Let n = (n1, . . . , nr) be a composition of n ∈ Z≥1, then Hilb ( 3T (0)(Kn1,...,nr) ab, t ) = ∑ k=(k1,...,kr) 0<kj≤nj (−t)\|n\|−\|k\| r∏ j=1 { nj kj } \|k\|−1∏ j=1 (1 + jt), where we set \|k\| := k1 + · · ·+ kr. Remark 4.20. This proposition is a consequence of Conjecture 4.15(1), which has been proved in [89]. Corollary 4.21. One has (a) 1 + t(t− 1) ∑ (n1,...,nr)∈Zr≥0\0r Hilb ( 3T (0)(Kn1,...,nr )ab , t) xn1 1 n1! · · · x nr r nr! = 1 + t r∑ j=1 (e−xj − 1) 1−t , (b) ∑ (n1,n2,...,nr)∈Z≥0\0r dim ( 3T (0)(Kn1,...,nr) ab )xn1 n1! · · · x nr nr! = − log 1− r + r∑ j=1 e−xj  , (c) Hilb ( 3T (0)(Kn1,...,nr) ab, t ) = (−t)\|n\|Chrom ( Kn1,...,nr ,−t−1 ) , (d) dim ( 3T (0)(Γ)ab ) is equal to the number of acyclic orientations of Γ, where Γ stands for a simple graph. Recall that for any graph Γ we denote by Chrom(Γ, x) the chromatic polynomial of that graph. Indeed, one can show36 Proposition 4.22. If r ∈ Z≥1, then Chrom(Kn1,...,nr , t) = ∑ k=(k1,...,kr) r∏ j=1 { nj kj } (t)\|k\|, where by definition (t)m := m−1∏ j=1 (t− j), (t)0 = 1, (t)m = 0 if m < 0. Finally we describe explicitly the exponential generating function for the Tutte polynomials of the weighted complete multipartite graphs. We refer the reader to [98] for a definition and a list of basic properties of the Tutte polynomial of a graph. 36If r = 1, the complete unipartite graph K(n) consists of n distinct points, and Chrom(K(n), x) = xn = n−1∑ k=0 { n k } (x)k. Let us stress that to abuse of notation the complete unipartite graph K(n) consists of n disjoint points with the Tutte polynomial equals to 1 for all n ≥ 1, whereas the complete graph Kn is equal to the complete multipartite graph K(1n). 62 A.N. Kirillov Definition 4.23. Let r ≥ 2 be a positive integer and {S1, . . . , Sr} be a collection of sets of cardinalities #\|Sj \| = nj , j = 1, . . . , r. Let ` := {`ij}1≤i<j≤n be a collection of non-negative integers. The `-weighted complete multipartite graph K (`) n1,...,nr is a graph with the set of vertices equals to the disjoint union r∐ j=1 Si of the sets S1, . . . , Sr, and the set of edges {(αi, βj), αi ∈ Si, βj ∈ Sj}1≤i<j≤r of multiplicity `ij each edge (αi, βj). Theorem 4.24. Let us fix an integer r ≥ 2 and a collection of non-negative integers ` := {`ij}1≤i<j≤r. Then 1 + ∑ n=(n1,...,nr)∈Zr≥0 n6=0 (x− 1)κ(`,n)Tutte ( K(`) n1,...,nr , x, y ) tn1 1 n1! · · · t nr r nr! =  ∑ m=(m1,...,mr)∈Zr≥0 y ∑ 1≤i<j≤r `ijmimj (y − 1)−\|m\| tm1 1 m1! · · · t mr r mr! (x−1)(y−1) , where κ(`,n) denotes the number of connected components of the graph K (`) n1,...,nr . Comments 4.25. (a) Clearly the condition `ij = 0 means that there are no edges between vertices from the sets Si and Sj . Therefore Theorem 4.24 allows to compute the Tutte polynomial of any (finite) graph. For example, Tutte ( K (16) 2,2,2,2, x, y ) = { (0, 362, 927, 911, 451, 121, 17, 1)x, (362, 2154, 2928, 1584, 374, 32)x, (1589, 4731, 3744, 1072, 96)x, (3376, 6096, 2928, 448, 16)x, (4828, 5736, 1764, 152)x, (5404, 4464, 900, 32)x, (5140, 3040, 380)x, (4340, 1840, 124)x, (3325, 984, 24)x, (2331, 448)x, (1492, 168)x, (868, 48)x, (454, 8)x, 210, 84, 28, 7, 1 } y . (b) One can show that a formula for the chromatic polynomials from Proposition 4.19 cor- responds to the specialization y = 0 (but not direct substitution!) of the formula for generating function for the Tutte polynomials stated in Theorem 4.24. (c) The Tutte polynomial Tutte ( K (`) n1,...,nr , x, y ) does not symmetric with respect to parameters {`ij}1≤i<j≤r. For example, let us write ` = (`12, `23, `13, `14, `24, `34), then Tutte ( K (6,3,4,5,2,4) 2,2,2,2 , 1, 1 ) = 28 · 3 · 5 · 113 · 241 = 1231760640. On the other hand, Tutte ( K (6,4,3,5,2,4) 2,2,2,2 , 1, 1 ) = 213 · 3 · 7 · 112 · 61 = 1269768192. 4.1.3 Universal Tutte polynomials Let m = (mij , 1 ≤ i < j ≤ n) be a collection of non-negative integers. Define generalized Tutte polynomial T̃n(m, x, y) as follows (x− 1)κ(n,m)T̃n(m, x, y) On Some Quadratic Algebras 63 = Coeff [t1···tn]   ∑ `1,...,`n `i∈{0,1}, ∀ i y ∑ 1≤i<j≤n mij`i`j (y − 1)−( ∑ j `j)t`11 · · · t `n n  (x−1)(y−1)  , where as before, κ(n,m) denotes the number of connected components of the graph K (m) n . Clearly that if Γ ⊂ K(`) n is a subgraph of the weighted complete graph K (`) n def = K (`) 1n , then the Tutte polynomial of graph Γ multiplied by (x− 1)κ(Γ) is equal to the following specialization mij = 0 if edge (i, j) /∈ Γ, mij = `ij if edge (i, j) ∈ Γ of the generalized Tutte polynomial (x− 1)κ(Γ)Tutte(Γ, x, y) = T̃n(m, x, y) ∣∣∣ mij=0 if (i,j)/∈Γ mij=`ij if (i,j)∈Γ . For example, (a) Take n = 6 and Γ = K6\{15, 16, 24, 25, 34, 36}, then Tutte(Γ, x, y) = {(0, 4, 9, 8, 4, 1)x, (4, 13, 9)x, (8, 7)x, 5, 1}y. (b) Take n = 6 and Γ = K6\{15, 26, 34}, then Tutte(Γ, x, y) = {(0, 11, 25, 20, 7, 1)x, (11, 46, 39, 8)x, (32, 52, 12)x, (40, 24)x, (29, 6)x, 15, 5, 1}y. (c) Take n = 6 and Γ = K6\{12.34.56} = K2,2,2. As a result one obtains an expression for the Tutte polynomial of the graph K2,2,2 displayed in Example 4.16(4). Now set us set qij := ymij − 1 y − 1 . Lemma 4.26. The generalized Tutte polynomial T̃n(m, x, y) is a polynomial in the variables {qij}1≤i<j≤n, x and y. Definition 4.27. The universal Tutte polynomial Tn({qij}, x, y) is defined to be the polynomial in the variables {qij}, x, and y defined in Lemma 4.26. Explicitly, (x− 1)Tn({qij}, x, y) = Coeff [t1···tn] ( ∑ `1,...,`n `i∈{0,1}, ∀ i ∏ 1≤i<j≤n (qij(y − 1) + 1)`i`j (y − 1)−( ∑ j `j)t`11 · · · t `n n )(x−1)(y−1)  . Corollary 4.28. Let {mij}1≤i<j≤n be a collection of positive integers. Then the specialization qij −→ [mij ]y := ymij − 1 y − 1 of the universal Tutte polynomial Tn({qij}, x, y) is equal to the Tutte polynomial of the complete graph Kn with each edge (i, j) of the multiplicity mij. 64 A.N. Kirillov Further specialization qij −→ 0 if edge (i, j) /∈ Γ allows to compute the Tutte polynomial for any graph Tutte3({q12, q13, q23}, x, y) = (1− q[12])(1− q[13])(1− q[23]) + yq[12]q[13]q[23]) + x(q[12] + q[13] + q[23]− 2) + x2. It is not difficult to see that the Tutten({qij}, x, y) is a symmetric polynomial with respect to parameters {qij}1≤i<j≤n. For more compact expression, it is more convenient to rewrite the universal chromatic poly- nomial in terms of parameters pij := 1− qij , 1 ≤ i < j ≤ n, and denote it by Chn({pij}, x). For example, Ch4({pij}, x) = −p12p13p14p23p24p34 + x ( 2− p12 − p13 − p14 − p23 − p24 − p34 + p12p34 + p14p23 + p13p24 + p12p13p23 + p12p14p24 + p13p14p34 + p23p24p34 ) + x2 ( 3− p12 − p13 − p14 − p23 − p24 − p34 ) + x3. Note that p12p34 + p14p23 + p13p24 is a symmetric polynomial of the variables p12, p34, p13, p24, p14, p23. It is important to keep in mind that parameters {mij} and {pij} are connected by relations pij = y − ymij y − 1 , 1 ≤ i < j ≤ n. Therefore, pij = 1 if (i, j) /∈ Edge(Γ), pij = 0 if mij = 1. We emphasize that the latter equalities are valid for arbitrary y. It is not difficult to see that Chn({qij = 0, ∀ i, j} = Tutte(Kn;x, 0), Chn({qij = 1, ∀ i, j} = (x− 1)n−1. Define universal chromatic polynomial to be Chn({pij}, x) = Tutten({pij}, x, 0), where we treat {pij}1≤i<j≤n as a collection of a free parameters. To state our result concerning the universal chromatic polynomial Chn({pij}, x), first we introduce a bit of notation. Let n ≥ 2 be an integer, consider a partition B = { Bi =( b (i) 1 , . . . , b (i) ri )} 1≤i≤k of the set [1, n] := [1, 2, . . . , n]. In other words one has that [1, n] = ∪ki=1Bi and Bi ∩ Bj = ∅ if i 6= j. We assume that b (1) 1 < b (2) 1 < · · · < b (k) 1 . We define κ(B) := k. To a given partition B we associate a monomial pB := k∏ a=1 pBa , where pBa = 1 if κ(B) = 1, and pBa = ∏ i,j∈Ba i<j pij . For a given partition λ ` n denote by L(β) λ ({pij}) the sum of all monomials pBβ κ(B)−2 such that λ = λ(B) def = (\|B1\|, . . . , \|Bκ(B)\|)+, where for any composition α \|= n, α+ denotes a unique partition obtained from α by the reordering of its parts. Define β-universal chromatic polynomial to be Ch(β) n ({pij}, x) = β−1L(n) + ∑ λ`n Tutte(K`(λ)−1;x, 0)L(β) λ , On Some Quadratic Algebras 65 where summation runs over all partitions λ of n; we set K0 := ∅ and Tutte(∅;x, y) = 0. For the reader convenience we are reminded that for the complete graph Kn, n > 0, one has Tutte(Kn, x, 0) = n−1∏ j=1 (x+ j − 1) = n−1∑ k=0 s(k, n− 1)xk, where s(k, n) denotes the Stirling number of the first kind37. Theorem 4.29 (formula for universal chromatic polynomials). Chn({pi,j}, x) = Ch(β=−1) n ({pij}, x). For a given partition λ ` n denote by Lλ({pij}) the sum of all monomials pB such that λ = λ(B) def = (\|B1\|, . . . , \|Bκ(B)\|)+, where for any composition α \|= n, α+ denotes a unique partition obtained from α by the reordering of its parts. It is clear that for a graph Γ ⊂ Kn and partition B the value of monomial pB under the specialization pij = 0 if (ij) ∈ Edge(Γ) and pij = 1 if (ij) /∈ Edge(Γ), is equal to 1 iff the complementary graph Kn\Γ contains a subgraph which is isomorphic to the disjoint union of complete graphs K(λ) := k∐ i=1 Kλi , where (λ1, . . . , λk) = λ(B). Therefore the specialization Lλ ∣∣ pij=0, (ij)∈Γ, pij=1, (ij)/∈Γ is equal to the number of non isomorphic subgraphs of the complementary graph Kn\Γ which are isomorphic to the graph K(λ). Example 4.30. Take n = 6, then Ch (β) 6 = β−1L(6) + x(x+ 1)(x+ 2)(x+ 3)(x+ 4)L(16)β 4 + x(x+ 1)(x+ 2)(x+ 3)L(2,14)β 3 + x(x+ 1)(x+ 2) ( L(22,12) + L(3,13) ) β2 + x(x+ 1) ( L(23) + L(3,2,1) + L(4,12) ) β + L(32) + L(4,2) + L(5,1). Since pij is equal to either 1 or 0, one can see that L(n) = 0 unless graph Γ is a collection of n distinct points and therefore L = 1. The chromatic polynomial of any graph is a Z-linear combination of the chromatic polyno- mials corresponding to a set of complete graphs. Corollary 4.31 (formula for universal β-Tutte polynomials38). (1− y)n−1Tutte(β) n ({pi,j};x, y) = ∏ 1≤i<j≤n pij + ∑ λ`n Tutte(K`(λ)−1;x+ y + βxy, 0)L(β) λ ({pij}). The polynomial (1 − y)\|V (Γ)\|−1Tutte(Γ;x, y) is a Z[y]-linear combination of the chromatic polynomials Tutte(Km;x+ y − xy, 0) corresponding to a family of complete graphs {Km}. Here V (Γ) denotes the set of vertices of graph Γ. 37See, e.g., [131, A008275] or https://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind. 38It should be remembered that Tutte(K1;x, y) = 1 and Tutte(K0;x, y) = 0, since the graph K1 := {pt} and graph K0 = ∅. https://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind 66 A.N. Kirillov Comments 4.32. (i) Let us write Ch(β) n ({pij}, x) = −L(n)β −1 + n−1∑ k=1 a(k) n ({pij)xk. It follows from Theorem 4.29 that a(k) n = ∑ λ`n s(`(λ)− 1, k)L(β) λ , where as before, s(k, n) denote the Stirling numbers of the first kind, see, e.g., footnote 37. For example, a(Γ) = a(1) n ({pij}) ∣∣∣∣ pij=0, (ij)∈Γ, pij=1, (ij)/∈Γ (`(λ)− 2)!Nλ(Γ)β`(λ)−2, where Nλ(Γ) denotes the number non isomorphic subgraphs in the complementary graph Kn\Γ, which are isomorphic to the graph Kn(λ). (ii) It is clear that for a general set of parameters {pij} the number of different monomials which appear in L(β) λ ({pij}), where partition λ = n∑ j=1 jmj , λ ` n, is equal to n!∏ j≥1 (j!)mjmj ! . (iii) For general set of parameters {pij} one can show that the number of different monomials which appear in polynomial a (1) n ({pij}) is equal to Bell(n)− 1, where Bell(n) denotes the n-th Bell number, see, e.g., [131, A000110]. (iv) In the limit y −→ 1 one has qij = mij and pij = 1−mij . (v) Let us introduce a modified universal Tutte polynomial, namely, Tutte({qij};x, y, z) := (−1)n−1Coeff [t1···tn] × [( ∑ `1,...,`n `i∈{0,1}, ∀ i ∏ 1≤i<j≤n (zqijy + 1)`i`jy−( ∑ j `j)t`11 · · · t `n n )xy x−1 ] . We set deg(qij) = 1, Proposition 4.33. (a) Tutte({qij};x, y, z) ∈ N[{qij ][x, y, z]. (b) Degree n−1 monomials of the polynomial Tutte({qij}; 0, y, z) are in one-to-one correspon- dence with the set of spanning trees of the complete graph Kn. Moreover, the polynomial Tutte({qij = 1, ∀ i, j};x, 0, 1) is equal to the generating function of forests on n labeled ver- tices, counting according to the number of connected components, whereas the polynomial Tutte({qij = 1, ∀ i, j}; 1, 0, z) is equal to the Hilbert polynomial of the even Orlik–Solomon On Some Quadratic Algebras 67 algebra39 OS+(Γn) associated to the type An−1 generic hyperplane arrangement Γn, see [119, Section 5] or [72], namely, Tutte({qij = 1, ∀ i, j}; 1, 0, z) = Hilb(OS+(Γn), z) = ∑ F z\|F\|, where the sum runs over all forests on the vertices {1, . . . , n}, and \|F\| denotes the number of edges of F . (c) More generally, denote by Fn(x, t) := ∑ F x\|F\|tinv(F) the generating function of statistics \|F\| and inv(F) on the set F (n) of forests on n labeled vertices. Recall that the symbol \|F\| denotes the number of edges in a forest F ∈ F (n) and that inv(F) its inversion index40. Lemma 4.34. One can show that Fn(x, t) = (xt)n−1Tutte(Kn; 1 + (xt)−1, t− 1), Coeff(xt)n−1 [Fn(x, t)] = In(t), where In(t) := ∑ F∈Tree(n) t inv(F) denotes the tree inversion polynomial, see, e.g., [51, 134]. (d) Set DUn(x) := (zt)n−1Hilb ( Kn; 1 + (zt)−1, z − 1 )∣∣∣ t:=−1 z:=−x = Fn(−x,−1). One (A.K.) can show that (n ≥ 2) DUn(x) ∈ N[x], DUn(1) = UDn+1, Coeffxn−1 [DUn(x)] = UDn−1, where UDn denote the Euler or up/down numbers associated with the exponential genera- ting function sec(x) + tan(x), see41, e.g., [131, A000111]. (e) One has x(n2)Tutte ( {qij = 1, ∀ i, j};x, x−1 − 1, 1 ) = Hilb(An, x), where An denotes the algebra generated by the curvature of 2-forms of the standard Her- mitian linear bundles over the flag variety F ln, see [72, 118, 129] or Section 4.2.2, Theo- rem 4.56(B). (f) Write Tutte({qij}; 0, y, z) = n−1∑ k=0 a (k) n (y, z), then monomials which appear in polynomial a (k) n (y, z) are in one-to-one correspondence with the set of labeled graphs with n nodes having exactly k connected components. (g) One has Tutte(({qij};x,−1, 1) = Tutte({qij}, x+ 1, 0). 39Known also as Orlik–Terao algebra. 40For the readers convenience we recall definitions of statistics inv(F) and the major index maj(F). Given a forest F on n labeled nodes, one can construct a tree T by adding a new vertex (root) connected with the maximal vertices in the connected components of F . The inversion index inv(F) is equal to the number of pairs (i, j) such that 1 ≤ i < j ≤ n, and the vertex labeled by j lies on the shortest path in T from the vertex labeled by i to the root. The major index maj(F) is equal to ∑ x∈Des(F) h(x); here for any vertex x ∈ F , h(x) is the size of the subtree rooted at x; the descent set Des(F) of F consists of the vertices x ∈ F which have the labeling strictly greater than the labeling of its child’s. 41The fact that In(−1) = UDn−1 is due to G. Kreweras [82]. 68 A.N. Kirillov (h) Recurrence relations for polynomials Fn(x, t), cf. [82], F0(x, t) = F1(x, t) = 1, Fn+1(x, t) = n∑ k=0 ( n k ) (xt)kIk(t)Fn−k(x, t). Example 4.35. Take n = 5, then Tutte(K5;x, y) = (0, 6, 11, 6, 1) + (6, 20, 10)y + 15(1, 1)y2 + 5(3, 1)y3 + 10y4 + 4y5 + y6, F5(−x,−1) = (1, 10, 25, 20, 5). Write Fn(x, t) = F̃n(u, t) ∣∣ u=xt , then F̃5(u, t) = 1 + 10u+ u2(35 + 10t) + u3(50, 40, 15, 5)t + u4(24, 36, 30, 20, 10, 4, 1)t, F̃n(u, 0) = n−1∏ j=1 (1 + j u), Hilb(A5, t) = (1, 4, 10, 20, 35, 51, 64, 60, 35, 10, 1)t, Hilb(OS+(Γ5), t) = (1, 10, 45, 110, 125)t. Exercises 4.36. (1) Assume that `ij = ` for all 1 ≤ i < j ≤ r. Based on the above formula for the exponential generating function for the Tutte polynomials of the complete multipartite graphsKn1,...,nr , deduce the following well-known formula Tutte ( K(`) n1,...,nr , 1, 1 ) = `N−1N r−2 r∏ j=1 (N − nj)nj−1, where N := n1 + · · ·+ nr. It is well-known that the number Tutte(Γ, 1, 1) is equal to the number of spanning trees of a connected graph Γ. (2) Take r = 3 and let n1, n2, n3 and `12, `13, `23 be positive integers. Set N := `12`13n1 + `12`23n2 + `13`23n3. Show that Tutte ( K`1,`2,`3 n1,n2,n3 , 1, 1 ) = N(`12n2+ `13n3)n1−1(`12n1+ `13n3)n2−1(`13n1+ `23n2)n3−1. (3) Let r ≥ 2, consider weighted complete multipartite graph K (`) n, . . . , n︸︷︷︸ r , where ` = (`ij) such that `1,j = `, j = 1, . . . , r and `ij = k, 2 ≤ i < j ≤ r. Show that Tutte ( K (`) n, . . . , n︸︷︷︸ r , 1, 1 ) = kn(r − 1)n−1((r − 1)`+ k)r−2((r − 2)`+ k)(r−1)(n−1)nnr−1. Let Γn(∗) be a spanning star subgraph of the complete graph Kn. For example, one can take for a graph Γn(∗) the subgraph K1,n−1 with the set of vertices V := {1, 2, . . . , n} and that of edges E := {(i, n), i = 1, . . . , n − 1}. The algebra 3T (0) n (K1,n−1) can be treated as a “noncommutative analog” of the projective space Pn−1. We have θ1 = u12 + u13 + · · · + u1n. It is not difficult to see that Hilb(3T (0) n (K1,n−1)ab, t) = (1 + t)n−1, and θn1 = 0. Let us observe that Chrom(Γn(?), t) = t(t− 1)n−1. Problem 4.37. Compute the Hilbert series of the algebra 3T (0) n (Kn1,...,nr). On Some Quadratic Algebras 69 The first non-trivial case is that of projective space, i.e., the case r = 2, n1 = 1, n2 = 5. On the other hand, if Γn = {(1, 2) → (2, 3) → · · · → (n − 1, n)} is the Dynkin graph of type An−1, then the algebra 3T (0) n (Γn) is isomorphic to the nil-Coxeter algebra of type An−1, and if Γ (aff) n = {(1, 2)→ (2, 3)→ · · · → (n− 1, n)→ −(1, n)} is the Dynkin graph of type A (1) n−1, i.e., a cycle, then the algebra 3T (0) n (Γ (aff) n ) is isomorphic to a certain quotient of the affine nil-Coxeter algebra of type A (1) n−1 by the two-sided ideal which can be described explicitly [72]. Moreover [72], Hilb ( 3T 0) n ( Γ(aff) ) , t ) = [n]t n−1∏ j=1 [j(n− j)]t, see Theorem 4.3. Therefore, the dimension dim(3T (0)(Γaff)) is equal to n!(n − 1)! and is equal also, as it was pointed out in Section 4.1.1, to the number of (directed) Hamiltonian cycles in the complete bipartite graph Kn,n, see [131, A010790]. It is not difficult to see that Hilb ( 3T (0) n (Γn)ab, t ) = (t+ 1)n−1, Hilb ( 3T (0) ( Γaff n )ab , t ) = t−1 ( (t+ 1)n − t− 1 ) , whereas Chrom(Γn, t) = t(t− 1)n−1, Chrom ( Γaff n , t ) = (t− 1)n + (−1)n(t− 1). Exercise 4.38. Let Kn1,...,nr be complete multipartite graph, N := n1 + · · ·+ nr. Show that42 Hilb(3TN (Kn1,...,nr), t) = r∏ j=1 nj−1∏ a=1 (1− at) N−1∏ j=1 (1− jt) . 4.1.4 Quasi-classical and associative classical Yang–Baxter algebras of type Bn In this section we introduce an analogue of the algebra 3Tn(β) for the classical root systems. Definition 4.39. (A) The quasi-classical Yang–Baxter algebra ̂ACYB(Bn) of type Bn is an associative algebra with the set of generators {xij , yij , zi, 1 ≤ i 6= j ≤ n} subject to the set of defining relations (1) xij + xij = 0, yij = yji if i 6= j, (2) zizj = zjzi, (3) xijxkl = xklxij , xijykl = yklxij , yijykl = yklyij if i, j, k, l are distinct, (4) zixkl = xklzi, ziykl = yklzi if i 6= k, l, (5) three term relations: xijxjk = xikxij + xjkxik − βxik, xijyjk = yikxij + yjkyik − βyik, xikyjk = yjkyij + yijxik + βyij , yikxjk = xjkyij + yijyik + βyij if 1 ≤ i < j < k ≤ n, 42It should be remembered that to abuse of notation, the complete graph Kn, by definition, is equal to the complete multipartite graph K((1, . . . , 1)︸︷︷︸ n ), whereas the graph K(n) is a collection of n distinct points. 70 A.N. Kirillov (6) four term relations: xijzj = zixij + yijzi + zjyij − βzi if i < j. (B) The associative classical Yang–Baxter algebra ACYB(Bn) of type Bn is the special case β = 0 of the algebra ̂ACYB(Bn). Comments 4.40. • In the case β = 0 the algebra ACYB(Bn) has a rational representation xij −→ (xi − xj)−1, yij −→ (xi + xj) −1, zi −→ x−1 i . • In the case β = 1 the algebra ̂ACYB(Bn) has a “trigonometric” representation xij −→ ( 1− qxi−xj )−1 , yij −→ ( 1− qxi+xj )−1 , zi −→ ( 1 + qxi )( 1− qxi )−1 . Definition 4.41. The bracket algebra E(Bn) of type Bn is an associative algebra with the set of generators {xij , yij , zi, 1 ≤ i 6= j ≤ n} subject to the set of relations (1)–(6) listed in Definition 4.39, and the additional relations (5a) xjkxij = xijxik + xikxjk − βxik, yjkxij = xijyik + yikyjk − βyik, yjkxik = yijyjk + xikyij + βyij , xjkyik = yijxjk + yikyij + βyij if 1 ≤ i < j < k ≤ n, (6a) zjxij = xijzi + ziyij + yijzj − βzi if i < j. Definition 4.42. The quasi-classical Yang–Baxter algebra ̂ACYB(Dn) of type Dn, as well as the algebras ACYB(Dn) and E(Dn) are defined by putting zi = 0, i = 1, . . . , n, in the corresponding Bn-versions of algebras in question. Conjecture 4.43. The both algebras E(Bn) and E(Dn) are Koszul, and Hilb(E(Bn), t) =  n∏ j=1 (1− (2j − 1)t) −1 ; if n ≥ 4 Hilb(E(Dn), t) = n−1∏ j=1 (1− 2jt) −1 . Example 4.44. Hilb(ACYB(B2), t) = ( 1− 4t+ 2t2 )−1 , Hilb(ACYB(B3), t) = ( 1− 9t+ 16t2 − 4t3 )−1 , Hilb(ACYB(B4), t) = ( 1− 16t+ 64t2 − 60t3 + 9t4 )−1 , Hilb(ACYB(D4), t) = ( 1− 12t+ 18t2 − 4t3 )−1 . However, Hilb(ACYB(B5), t) = ( 1− 25t+ 180t2 − 400t3 + 221t4 − 31t5 )−1 . On Some Quadratic Algebras 71 Let us introduce the following Coxeter type elements hBn := n−1∏ a=1 xa,a+1zn ∈ E(Bn) and hDn := n−1∏ a=1 xa,a+1yn−1,n ∈ E(Dn). (4.2) Let us bring the element hBn (resp. hDn) to the reduced form in the algebra E(Bn) that is, let us consecutively apply the defining relations (1)–(6), (5a), (6a) to the element hBn (resp. apply to hDn the defining relations for algebra E(Dn)) in any order until unable to do so. Denote the resulting (noncommutative) polynomial by PBn(xij , yij , z) (resp. PDn(xij , yij)). In principal, this polynomial itself can depend on the order in which the relations (1)–(6), (5a), (6a) are applied. Conjecture 4.45 (cf. [133, Exercise 8.C5(c)]). (1) Apart from applying the commutativity relations (1)–(4), the polynomial PBn(xij , yij , z) (resp. PDn(xij , yij)) does not depend on the order in which the defining relations have been applied. (2) Define polynomial PBn(s, r, t) (resp. PDn(s, r)) to be the the image of that PBn(xij , yij , z) (resp. PDn(xij , yij)) under the specialization xij −→ s, yij −→ r, zi −→ t. Then PBn(1, 1, 1) = 1 2 ( 2n n ) = 1 2CatBn. Note that PBn(1, 0, 1) = CatAn−1 . Problem 4.46. Investigate the Bn and Dn types reduced polynomials corresponding to the Co- xeter elements (4.2), and the reduced polynomials corresponding to the longest elements wBn := n∏ J=1 zj  ∏ 1≤i<j≤n xijyij  , wDn = ∏ 1≤i<j≤n xijyij . 4.2 Super analogue of 6-term relations and classical Yang–Baxter algebras 4.2.1 Six term relations algebra 6Tn, its quadratic dual (6Tn)!, and algebra 6HTn Definition 4.47. The 6 term relations algebra 6Tn is an associative algebra (say over Q) with the set of generators {ri,j , 1 ≤ i 6= j < n}, subject to the following relations: 1) ri,j and rk,l commute if {i, j} ∩ {k, l} = ∅, 2) unitarity condition: rij + rji = 0, 3) classical Yang–Baxter relations: [rij , rik + rjk] + [rik, rjk] = 0 if i, j, k are distinct. We denote by CYBn, named by classical Yang–Baxter algebra, an associative algebra over Q generated by elements {rij , 1 ≤ i 6= j ≤ n} subject to relations 1) and 3). Note that the algebra 6Tn is given by ( n 2 ) generators and ( n 3 ) + 3 ( n 4 ) quadratic relations. Definition 4.48. Define Dunkl elements in the algebra 6Tn to be θi = ∑ j 6=i rij , i = 1, . . . , n. It easy to see that the Dunkl elements {θi}1≤i≤n generate a commutative subalgebra in the algebra 6Tn. 72 A.N. Kirillov Example 4.49 (some “rational and trigonometric” representations of the algebra 6Tn). Let A = U(sl(2)) be the universal enveloping algebra of the Lie algebra sl(2). Recall that the algebra sl(2) is spanned by the elements e, f , h, such that [h, e] = 2e, [h, f ] = −2f , [e, f ] = h. Let’s search for solutions to the CYBE in the form ri,j = a(ui, uj)h⊗ h+ b(ui, uj)e⊗ f + c(ui, uj)f ⊗ e, where a(u, v), b(u, v) 6= 0, c(u, v) 6= 0 are meromorphic functions of the variables (u, v) ∈ C2, defined in a neighborhood of (0, 0), taking values in A ⊗ A. Let aij := a(ui, uj) (resp. bij := b(ui, uj), cij := c(ui, uj)). Lemma 4.50. The elements ri,j := aijh⊗ h+ bije⊗ f + cijf ⊗ e satisfy CYBE iff bijbjkcik = cijcjkbik and 4aik = bijbjk/bik − bikcjk/bij − bikcij/bjk for 1 ≤ i < j < k ≤ n. It is not hard to see that • there are three rational solutions: r1(u, v) = 1/2h⊗ h+ e⊗ f + f ⊗ e u− v , r2(u, v) = u+ v 4(u− v) h⊗ h+ u u− v e⊗ f + v u− v f ⊗ e, and r3(u, v) := −r2(v, u), • there is a trigonometric solution rtrig(u, v) = 1 4 q2u + q2v q2u − q2v h⊗ h+ qu+v q2u − q2v ( e⊗ f + f ⊗ e ) . Notice that the Dunkl element θj := ∑ a6=j rtrig(ua, uj) corresponds to the truncated (or level 0) trigonometric Knizhnik–Zamolodchikov operator. In fact, the “sln-Casimir element” Ω = 1 2 ( n∑ i=1 Eii ⊗ Eii ) + ∑ 1≤i<j≤n Eij ⊗ Eji satisfies the 4-term relations [Ω12,Ω13 + Ω23] = 0 = [Ω12 + Ω13,Ω23], and the elements rij := Ωij ui−uj , 1 ≤ i < j ≤ n, satisfy the classical Yang–Baxter relations. Recall that the set {Eij := (δikδjl)1≤k,l≤n, 1 ≤ i, j ≤ n}, stands for the standard basis of the algebra Mat(n,R). Definition 4.51. Denote by 6T (0) n the quotient of the algebra 6Tn by the (two-sided) ideal generated by the set of elements {r2 i,j , 1 ≤ i < j ≤ n}. More generally, let {β, qij , 1 ≤ i < j ≤ n} be a set of parameters. Let R := Q[β][q±1 ij ]. Definition 4.52. Denote by 6HTn the quotient of the algebra 6Tn⊗R by the (two-sided) ideal generated by the set of elements {r2 i,j − βri,j − qij , 1 ≤ i < j ≤ n}. On Some Quadratic Algebras 73 All these algebras are naturally graded, with deg(ri,j) = 1, deg(β) = 1, deg(qij) = 2. It is clear that the algebra 6T (0) n can be considered as the infinitesimal deformation Ri,j := 1 + εri,j , ε −→ 0, of the Yang–Baxter group YBn. For the reader convenience we recall the definition of the Yang–Baxter group. Definition 4.53. The Yang–Baxter group YBn is a group generated by elements {R±1 ij , 1 ≤ i < j ≤ n}, subject to the set of defining relations • RijRkl = RklRij if i, j, k, l are distinct, • quantum Yang–Baxter relations: RijRikRjk = RjkRikRij if 1 ≤ i < j < k ≤ n. Corollary 4.54. Define hij = 1+rij ∈ 6HTn. Then the following relations in the algebra 6HTn are satisfied: (1) rijrikrjk = rjkrikrij for all pairwise distinct i, j and k; (2) Yang–Baxter relations: hijhikhjk = hjkhikhij if 1 ≤ i < j < k ≤ n. Note, the item (1) includes three relations in fact. Proposition 4.55. (1) The quadratic dual (6Tn)! of the algebra 6Tn is a quadratic algebra generated by the elements {ti,j , 1 ≤ i < j ≤ n} subject to the set of relations (i) t2i,j = 0 for all i 6= j; (ii) anticommutativity: tijtk,l + tk,lti,j = 0 for all i 6= j and k 6= l; (iii) ti,jti,k = ti,ktj,k = ti,jtj,k if i, j, k are distinct. (2) The quadratic dual (6T (0) n )! of the algebra 6T (0) n is a quadratic algebra with generators {ti,j , 1 ≤ i < j ≤ n} subject to the relations (ii)–(iii) above only. 4.2.2 Algebras 6T (0) n and 6TFn We are reminded that the algebra 6T (0) n is the quotient of the six term relation algebra 6Tn by the two-sided ideal generated by the elements {rij}1≤i<j≤n. Important consequence of the classical Yang–Baxter relations and relations r2 ij = 0, ∀ i 6= j, is that the both additive Dunkl elements {θi}1≤i≤n and multiplicative onesΘi = 1∏ a=i−1 h−1 ai n∏ a=i+1 hia  1≤i≤n generate commutative subalgebras in the algebra 6T (0) n (and in the algebra 6Tn as well), see Corollary 4.54. The problem we are interested in, is to describe commutative subalgebras gen- erated by additive (resp. multiplicative) Dunkl elements in the algebra 6T (0) n . Notice that the subalgebra generated by additive Dunkl elements in the abelianization43 of the algebra 6Tn(0) has been studied in [118, 129]. In order to state the result from [118] we need, let us introduce a bit of notation. As before, let F ln denotes the complete flag variety, and denote by An the algebra generated by the curvature of 2-forms of the standard Hermitian linear bundles over 43See, e.g., http://mathworld.wolfram.com/Abelianization.html. http://mathworld.wolfram.com/Abelianization.html 74 A.N. Kirillov the flag variety F ln, see, e.g., [118]. Finally, denote by In the ideal in the ring of polynomials Z[t1, . . . , tn] generated by the set of elements (ti1 + · · ·+ tik)k(n−k)+1 for all sequences of indices 1 ≤ i1 < i2 < · · · < ik ≤ n, k = 1, . . . , n. Theorem 4.56 ([118, 129]). (A) There exists a natural isomorphism An −→ Z[t1, . . . , tn]/In, (B) Hilb(An, t) = t( n 2)Tutte ( Kn, 1 + t, t−1 ) . Therefore the dimension of An (as a Z-vector space) is equal to the number F(n) of forests on n labeled vertices. It is well-known that ∑ n≥1 F(n) xn n! = exp ∑ n≥1 nn−1x n n! − 1. For example, Hilb(A3, t) = (1, 2, 3, 1), Hilb(A4, t) = (1, 3, 6, 10, 11, 6, 1), Hilb(A5, t) = (1, 4, 10, 20, 35, 51, 64, 60, 35, 10, 1), Hilb(A6, t) = (1, 5, 15, 35, 70, 126, 204, 300, 405, 490, 511, 424, 245, 85, 15, 1). Problem 4.57. Describe subalgebra in (6T (0) n )ab generated by the multiplicative Dunkl elements {Θi}1≤i≤n. On the other hand, the commutative subalgebra Bn generated by the additive Dunkl elements in the algebra 6T (0) n , n ≥ 3, has infinite dimension. For example, B3 ∼= Z[x, y]/〈xy(x+ y)〉, and the Dunkl elements θ (3) j , j = 1, 2, 3, have infinite order. Definition 4.58. Define algebra 6TFn to be the quotient of that 6T (0) n by the two-sided ideal generated by the set of “cyclic relations” m∑ j=2 m∏ a=j ri1,ia j∏ a=2 ri1,ia = 0 for all sequences {1 ≤ i1, i2, . . . , im ≤ n} of pairwise distinct integers, and all integers 2 ≤ m ≤ n. For example, • Hilb(6TF3 , t) = (1, 3, 5, 4, 1) = (1 + t)(1, 2, 3, 1), • subalgebra (over Z) in the algebra 6TF3 generated by Dunkl elements θ1 and θ2 has the Hilbert polynomial equal to (1, 2, 3, 1), and the following presentation: Z[x, y]/I3, where I3 denotes the ideal in Z[x, y] generated by x3, y3, and (x+ y)3, On Some Quadratic Algebras 75 • Hilb(6TF4 , t) = (1, 6, 23, 65, 134, 164, 111, 43, 11, 1)t. As a consequence of the cyclic relations, one can check that for any integer n ≥ 2 the n-th power of the additive Dunkl element θi is equal to zero in the algebra 6TFn for all i = 1, . . . , n. Therefore, the Dunkl elements generate a finite-dimensional commutative subalgebra in the algebra 6TFn . There exist natural homomorphisms 6TFn −→ 3T (0) n , Bn π̃−−−−→ An −→ H∗(F ln,Z) (4.3) The first and third arrows in (4.3) are epimorphism. We expect that the map π̃ is also epimor- phism44, and looking for a description of the kernel ker(π̃). Comments 4.59. • Let us denote by Bmult n and Amult n the subalgebras generated by multiplicative Dunkl ele- ments in the algebras 6T (0) n and ( 6T (0) n )ab correspondingly. One can define a sequence of maps Bmult n −→ Amult n φ̃−→ K∗(F ln), (4.4) which is a K-theoretic analog of that (4.3). It is an interesting problem to find a geometric interpretation of the algebra Amult n and the map φ̃. • “Quantization”. Let β and {qij = qji, 1 ≤ i, j ≤ n} be parameters. Definition 4.60. Define algebra 6HTn to be the quotient of the algebra 6Tn by the two sided ideal generated by the elements {r2 ij − βrij − qij}1≤i,j≤n. Lemma 4.61. The both additive {θi}1≤i≤n and multiplicative {Θi}1≤i≤n Dunkl elements gen- erate commutative subalgebras in the algebra 6HTn. Therefore one can define algebras 6HBn and 6HAn which are a “quantum deformation” of algebras Bn and An respectively. We expect that in the case β = 0 and a special choice of “arithmetic parameters” {qij}, the algebra HAn is connected with the arithmetic Schubert and Grothendieck calculi, cf. [129, 137]. Moreover, for a “general” set of parameters {qij}1≤i,j≤n and β = 0, we expect an existence of a natural homomorphism HAmult n −→ QK∗(F ln), where QK∗(F ln) denotes a multiparameter quantum deformation of the K-theory ring K∗(F ln) [72, 76]; see also Section 3.1. Thus, we treat the algebra HAmult n as the K-theory version of a multiparameter quantum deformation of the algebra Amult n which is generated by the curvature of 2-forms of the Hermitian linear bundles over the flag variety F ln. • One can define an analogue of the algebras 6T (0) n , 6HTn etc., denoted by 6T (Γ) etc., for any subgraph Γ ⊂ Kn of the complete graph Kn, and in fact for any oriented matroid. It is known that Hilb((6Tn(Γ)ab, t) = te(Γ)Tutte(Γ, 1 + t, t−1), see, e.g., [11] and the literature quoted therein. 44Contrary to the case of the map prn : Z[θ1, . . . , θn] −→ (3Tn(0))ab, where the image Im(prn) has dimension equals to the number of permutations in Sn with (n− 1) inversions see [131, A001892]. 76 A.N. Kirillov 4.2.3 Hilbert series of algebras CYBn and 6Tn 45 Examples 4.62. Hilb(6T3, t) = ( 1− 3t+ t2 )−1 , Hilb(6T4, t) = ( 1− 6t+ 7t2 − t3 )−1 , Hilb(6T5, t) = ( 1− 10t+ 25t2 − 15t3 + t4 )−1 , Hilb(6T6, t) = ( 1− 15t+ 65t2 − 90t3 + 31t4 − t5 )−1 , Hilb ( 6T (0) 3 , t ) = [2][3](1− t)−1, Hilb ( 6T (0) 4 , t ) = [4](1− t)−2 ( 1− 3t+ t2 )−1 . In fact, the following statements are true. Proposition 4.63 (cf. [7]). Let n ≥ 2, then • The algebras 6Tn and CYBn are Koszul. • We have Hilb(6Tn, t) = ( n−1∑ k=0 (−1)k { n n− k } tk )−1 , where { n k } stands for the Stirling numbers of the second kind, i.e., the number of ways to partition a set of n things into k nonempty subsets. • Hilb(CYBn, t) = ( n−1∑ k=0 (−1)k(k + 1)!N(k, n)tk )−1 , where N(k, n) = 1 n ( n k )( n k+1 ) denotes the Narayana number, i.e., the number of Dyck n-paths with exactly k peaks. Corollary 4.64. (A) The Hilbert polynomial of the quadratic dual of the algebra 6Tn is equal to Hilb ( 6T ! n, t ) = n−1∑ k=0 { n n− k } tk. It is well-known that∑ n≥0 ( n−1∑ k=0 { n n− k } tk ) zn n! = exp ( exp(zt)− 1 t ) . Therefore, dim(6Tn)! = Belln, where Belln denotes the n-th Bell number, i.e., the number of ways to partition n things into subsets, see [131]. Recall, that∑ n≥0 Belln zn n! = exp(exp(z)− 1)). 45Results of this subsection have been obtained independently in [7]. This paper contains, among other things, a description of a basis in the algebra 6Tn, and much more. On Some Quadratic Algebras 77 (B) The Hilbert polynomial of the quadratic dual of the algebra CYBn is equal to Hilb ( (CYBn )! , t) = n−1∑ k=0 (k + 1)!N(k, n)tk = (n− 1)!L (α=1) n−1 ( −t−1 ) tn−1, where L(α) n (x) = x−αex n! dn dxn ( e−xxn+α ) denotes the generalized Laguerre polynomial. The numbers (k + 1)!N(n, k) := L(n, n− k) are known as Lah numbers, see, e.g., [131, A008297], moreover [131], dim(CYBn)! = A000262. It is well-known that ∑ n≥0 n−1∑ k≥0 (k + 1)!N(k, n)tk  zn n! = exp ( z(1− zt)−1 ) . Comments 4.65. Let En(u), u 6= 0, 1, be the Yokonuma–Hecke algebra, see, e.g., [124] and the literature quoted therein. It is known that the dimension of the Yokonuma–Hecke algebra En(u) is equal to n!Bn, where Bn denotes as before the n-th Bell number. Therefore, dim(En(u)) = dim((6Tn)! o Sn), where (6Tn)! o Sn denotes the semi-direct product of the algebra (6Tn)! and the symmetric group Sn. It seems an interesting task to check whether or not the algebras (6Tn)! o Sn and En(u) are isomorphic. Remark 4.66. Denote by MYBn the group algebra over Q of the monoid corresponding to the Yang–Baxter group YBn, see, e.g., Definition 4.48. Let P (MYBn, s, t) denotes the Poincaré polynomial of the algebra MYBn. One can show that Hilb(6Tn, s) = P (MYBn,−s, 1)−1. For example, P (MYB3, s, t) = 1 + 3st+ s2t3, P (MYB4, s, t) = 1 + 6st+ s2 ( 3t2 + 4t3 ) + s3t6, P (MYB5, s, t) = 1 + 10st+ s2 ( 15t2 + 10t3 ) + s3 ( 10t4 + 5t6 ) + s4t10. Note that Hilb(MYBn, t) = P (MYBn,−1, t)−1 and P (MYBn, 1, 1) = Belln, the n-th Bell number. Conjecture 4.67. P (MYBn, s, t) = ∑ π s#(π)tn(π), where the sum runs over all partitions π = (I1, . . . , Ik) of the set [n] := [1, . . . , n] into nonempty subsets I1, . . . , Ik, and we set by definition, #(π) := n− k, n(π) := k∑ a=1 (\|Ia\| 2 ) . Remark 4.68. For any finite Coxeter group (W,S) one can define the algebra CYB(W ) := CYB(W,S) which is an analog of the algebra CYBn = CYB(An−1) for other root systems. 78 A.N. Kirillov Conjecture 4.69 (A.N. Kirillov, Yu. Bazlov). Let (W,S) be a finite Coxeter group with the root system Φ. Then • the algebra CYB(W ) is Koszul; • Hilb(CYB(W ), t) = { \|S\|∑ k=0 rk(Φ)(−t)k }−1 , where rk(Φ) is equal to the number of subsets in Φ+ which constitute the positive part of a root subsystem of rank k. For example, r1(Φ) = \|Φ+\|, and r2(Φ) is equal to the number of defining relations in a representation of the algebra CYB(W ). Example 4.70. Hilb ( CYB(B2)!, t ) = (1, 4, 3), Hilb ( CYB(B3)!, t ) = (1, 9, 13, 2), Hilb ( CYB(B4)!, t ) = (1, 16, 46, 28, 5), Hilb ( CYB(B5)!, t ) = (1, 25, 130, 200, 101, 12), Hilb ( CYB(D4)!, t ) = (1, 12, 34, 24, 4), Hilb ( CYB(D5)!, t ) = (1, 20, 110, 190, 96, 11). Definition 4.71. The even generic Orlik–Solomon algebra OS+(Γn) is defined to be an associa- tive algebra (say over Z) generated by the set of mutually commuting elements yi,j , 1 ≤ i 6= j ≤ n, subject to the set of cyclic relations yi,j = yj,i, yi1,i2yi2,i3 · · · yik−1,ikyi1,ik = 0 for k = 2, . . . , n, and all sequences of pairwise distinct integers 1 ≤ i1, . . . , ik ≤ n. Exercises 4.72. (1) Show that exp ( z(1− zt)−q ) = 1 + ∑ n≥1 ( 1 + n−1∑ k=1 ( n− 1 k ) k−1∏ a=0 (a+ (n− k)q)tk ) zn n! . (2) The even generic Orlik–Solomon algebra. Show that the number of degree k, k ≥ 3, relations in the definition of the Orlik–Solomon algebra OS+(Γn) is equal to 1 2(k − 1)! ( n k ) and also is equal to the maximal number of k-cycles in the complete graph Kn. Note that if one replaces the commutativity condition in the above definition on the condi- tion that yi,j ’s pairwise anticommute, then the resulting algebra appears to be isomorphic to the Orlik–Solomon algebra OS(Γn) corresponding to the generic hyperplane arrangement Γn, see [119]. It is known [119, Corollary 5.3], that Hilb(OS(Γn), t) = ∑ F t\|F \|, where the sum runs over all forests F on the vertices 1, . . . , n, and \|F \| denotes the number of edges in a forest F . It follows from Corollary 4.64, that∑ n≥1 Hilb(OS(Γn), t) zn n! = exp ∑ n≥1 nn−2tn−1 z n n!  . It is not difficult to see that Hilb(OS+(Γn), t) = Hilb(OS(Γn), t). In particular, dim OS+(Γn) = F(n). Note also that a sequence {Hilb(OS(Γn),−1)}n≥2 appears in [131, A057817]. The poly- nomials Hilb(An, t), Fn(x, t) and Hilb(OS+(Γn), t) can be expressed, see, e.g., [118], as certain specializations of the Tutte polynomial T (G;x, y) corresponding to the complete graph G := Kn. Namely, Hilb(An, t) = t( n 2)T ( Kn; 1 + t, t−1 ) , Hilb ( OS+(Γn), t ) = tn−1T ( Kn; 1 + t−1, 1 ) . On Some Quadratic Algebras 79 4.2.4 Super analogue of 6-term relations algebra Let n, m be non-negative integers. Definition 4.73. The super 6-term relations algebra 6Tn,m is an associative algebra over Q generated by the elements {xi,j , 1 ≤ i 6= j ≤ n} and {yα,β, 1 ≤ α 6= β ≤ m} subject to the set of relations (0) xi,j + xj,i = 0, yα,β = yβ,α; (1) xi,jxk,l = xk,lxi,j , xi,jyα,β = yα,βxi,j , yα,βyγ,δ + yγ,δyα,β = 0, if tuples (i, j, k, l), (i, j, α, β), as well as (α, β, γ, δ) consist of pair-wise distinct integers; (2) classical Yang–Baxter relations and theirs super analogue: [xi,k, xj,i+xj,k]+ [xi,j , xj,k] = 0 if 1 ≤ i, j, k ≤ n are distinct, [xi,k, yj,i + yj,k] + [xi,j , yj,k] = 0 if 1 ≤ i, j, k ≤ min(n,m) are distinct, [yα,γ , yβ,α + yβ,γ ]+ + [yα,β, yβ,γ ]+ = 0 if 1 ≤ α, β, γ ≤ m are distinct. Recall that [a, b]+ := ab+ ba denotes the anticommutator of elements a and b. Conjecture 4.74. The algebra 6Tn,m is Koszul. Theorem 4.75. Let n,m ∈ Z≥1, one has Hilb ( (6Tn)!, t ) Hilb ( (6Tm)!, t ) = min(n,m)−1∑ k=0 { min(n,m) min(n,m)− k } Hilb ( (6Tn−k,m−k) !, t ) t2k, where as before { n n−k } denotes the Stirling numbers of the second kind, see, e.g., [131, A008278]. Corollary 4.76. Let n,m ∈ Z≥1. One has (a) Symmetry: Hilb(6Tn,m, t) = Hilb(6Tm,n, t). (b) Let n ≤ m, then Hilb ( (6Tn,m)!, t ) = n−1∑ k=0 s(n− 1, n− k)Hilb ( (6Tn−k) !, t ) Hilb ( (6Tm−k) !, t ) t2k, where s(n− 1, n− k) denotes the Stirling numbers of the first kind, i.e., n−1∑ k=0 s(n− 1, n− k)tk = n−1∏ j=1 (1− jt). (c) dim(6Tn,n)! is equal to the number of pairs of partitions of the set {1, 2, . . . , n} whose meet is the partition {{1}, {2}, . . . , {n}}, see, e.g., [131, A059849]. Example 4.77. Hilb ( (6T3,2)!, t ) = Hilb ( (6T2,3)!, t ) = (1, 4, 3), Hilb ( (6T2,4)!, t ) = Hilb ( (6T4,2)!, t ) = (1, 7, 12, 5), Hilb ( (6T3,3)!, t ) = (1, 6, 8), Hilb ( (6T2,5)!, t ) = Hilb ( (6T5,2)!, t ) = (1, 11, 34, 34, 9), Hilb ( (6T3,4)!, t ) = Hilb ( (6T4,3)!, t ) = (1, 9, 23, 16), Hilb ( (6T4,4)!, t ) = (1, 12, 44, 50, 6), Hilb ( (6T3,5)!, t ) = Hilb ( (6T5,3)!, t ) = (1, 13, 53, 79, 34), Hilb ( (6T4,5)!, t ) = Hilb ( (6T5,4)!, t ) = (1, 16, 86, 182, 131, 12), Hilb ( (6T5,5)!, t ) = (1, 20, 140, 410, 462, 120). 80 A.N. Kirillov Now let us define in the algebra 6Tn,m the Dunkl elements θi := ∑ j 6=i xi,j , 1 ≤ i ≤ n, and θ̄α := ∑ β 6=α yα,β, 1 ≤ α ≤ m. Lemma 4.78. One has • [θi, θj ] = 0, • [θi, θ̄α] = [xi,α, yi,α], • [θ̄α, θ̄β]+ = 2y2 α,β if α 6= β. Remark 4.79 (“odd” six-term relations algebra). In particular, one can define an “odd” analog 6T (−) n = 6T0,n of the six term relations algebra 6Tn. Namely, the algebra 6T (−) n is given by the set of generators {yij , 1 ≤ i < j ≤ n}, and that of relations: 1) yi,j and yk,l anticommute if i, j, k, l are pairwise distinct; 2) [yi,j , yi,k + yj,k]+ + [yi,k, yj,k]+ = 0, if 1 ≤ i < j ≤ k ≤ n, where [x, y]+ = xy + yx denotes the anticommutator of x and y. The “odd” three term relations algebra 3T−n can be obtained as the quotient of the alge- bra 6T−n by the two-sided ideal generated by the three term relations yijyjk + yjkyki+ ykiyij = 0 if i, j, k are pairwise distinct. One can show that the Dunkl elements θi and θj , i 6= j, given by formula θi = ∑ j 6=i yij , i = 1, . . . , n, form an anticommutative family of elements in the algebra 6T (−) n . In a similar fashion one can define an “odd” analogue of the dynamical six term relations algebra 6DTn, see Definition 2.3 and Section 2.1.1, as well as define an “odd’ analogues of the algebra 3MTn(β,0), see Definition 3.7, the Kohno–Drinfeld algebra, the Hecke algebra and few others considered in the present paper. Details are omitted in the present paper. More generally, one can ask what are natural q-analogues of the six term and three term relations algebras? In other words to describe relations which ensure the q-commutativity of Dunkl elements defined above. First of all it would appear natural that the “q-locality and q- symmetry conditions” hold among the set of generators {yij , 1 ≤ i 6= j ≤ n}, that is yij + qyji = 0, yijykl = qyklyij if i < j, k < l, and {i, j} ∩ {k, l} = ∅. Another natural condition is the fulfillment of q-analogue of the classical Yang–Baxter rela- tions, namely, [yik, yjk]q+[yik, yji]q+[yij , yjk]q = 0 if i < j < k, where [x, y]q := xy−qyx denotes the q-commutator. However we are not able to find the q-analogue of the classical Yang–Baxter relation listed above in the mathematical and physical literature yet. Only cases q = 1 and q = −1 have been extensively studied. 4.3 Four term relations algebras / Kohno–Drinfeld algebras 4.3.1 Kohno–Drinfeld algebra 4Tn and algebra CYBn Definition 4.80. The 4-term relations algebra (or the Kohno–Drinfeld algebra, or infinitesimal pure braids algebra) 4Tn is an associative algebra (say over Q) with the set of generators yi,j , 1 ≤ i < j ≤ n, subject to the following relations 1) yi,j and yk,l are commute, if i, j, k, l are all distinct; 2) [yi,j , yi,k + yj,k] = 0, [yi,j + yi,k, yj,k] = 0 if 1 ≤ i < j ≤ k ≤ n. On Some Quadratic Algebras 81 Note that the algebra 4Tn is given by ( n 2 ) generators and 2 ( n 3 ) + 3 ( n 4 ) quadratic relations, and the element c := ∑ 1≤i<j≤n yi,j belongs to the center of the Kohno–Drinfeld algebra. Definition 4.81. Denote by 4T (0) n the quotient of the algebra 4Tn by the (two-sided) ideal generated by by the set of elements {y2 i,j , 1 ≤ i < j ≤ n}. More generally, let β, {qij , 1 ≤ i < j ≤ n} be the set of parameters, denote by 4HTn the quotient of the algebra 4Tn by the two-sided ideal generated by the set of elements {y2 ij −βyij − qij , 1 ≤ i < j ≤ n}. These algebras are naturally graded, with deg(yi,j) = 1, deg(β) = 1, deg(qij) = 2, as well as each of that algebras has a natural filtration by setting deg(yi,j) = 1, deg(β) = 0, deg(qij) = 0, ∀ i 6= j. It is clear that the algebra 4Tn can be considered as the infinitesimal deformation gi,j := 1 + εyi,j , ε −→ 0, of the pure braid group Pn. There is a natural action of the symmetric group Sn on the algebra 4Tn (and also on 4T 0 n) which preserves the grading: it is defined by w · yi,j = yw(i),w(j) for w ∈ Sn. The semi-direct product QSnn4Tn (and also that QSnn4T 0 n) is a Hopf algebra denoted by Bn (respectively B(0) n ). Remark 4.82. There exists the natural map CYBn −→ 4Tn given by yi,j := ui,j + uj,i. Indeed, one can easily check that [yij , yik + yjk] = wijk + wjik − wkij − wkji, see Section 2.3.1, Definition 2.21 for a definition of the classical Yang–Baxter algebra CY Bn, and Section 2, equation (2.3), for a definition of the element wijk. Remark 4.83. • Much as the relations in the algebra 6Tn are chosen in a way to imply (and “essentially”46 equivalent) the pair-wise commutativity of the Dunkl elements {θi}1≤i≤n, the relations in the Kohno–Drinfeld algebra imply (and “essentially” equivalent) to pair-wise commutati- vity of the Jucys–Murphy elements (or, equivalently, dual JM-elements) dj := ∑ 1≤a<j yaj , 2 ≤ j ≤ n (resp. di = ∑ 1≤a≤i yn−i,n−a+1, 1 ≤ i ≤ n− 1). • It follows from the classical 3-term identity (“Jacobi identity”) 1 (a− b)(a− c) − 1 (a− b)(b− c) + 1 (a− c)(b− c) = 0, that if elements {yi,j \| 1 ≤ i < j ≤ n} satisfy the 4-term algebra relations, see Defini- tion 4.80, and t1, . . . , tn, a set of (pair-wise) commuting parameters, then the elements ri,j := yi,j ti − tj 46Together with locality and factorization conditions a set of defining relations in the algebra 6Tn is equivalent to the commutativity property of Dunkl’s elements. 82 A.N. Kirillov satisfy the set of defining relations of the 6-term relations algebra 6Tn, see Section 4.2.1, Definition 4.47. In particular, the Knizhnik–Zamolodchikov elements KZj := ∑ i 6=j yi,j ti − tj , 1 ≤ j ≤ n, form a pair-wise commuting family (by definition, we put yi,j = yj,i if i > j). Example 4.84. (1) Yang representation of the 4Tn. Let Sn be the symmetric group acting identically on the set of variables {x1, . . . , xn}. Clearly that the elements {yi,j := sij}1≤i<j≤n, yi,j := yj,i if i > j, satisfy the Kohno–Drinfeld relations listed in Definition 4.80. Therefore the operators uij defined by uij = (xi − xj)−1sij give rise to a representation of the algebra 3Tn on the field of rational functions Q(x1, . . ., xn). The Dunkl–Gaudin elements θi = ∑ j,j 6=i yij , i = 1, . . . , n correspond to the truncated Gaudin operators acting in the tensor space (C)⊗n. Cf. Section 3.3. (2) Let A = U(sl(2)) be the universal enveloping algebra of the Lie algebra sl(2). Recall that the algebra sl(2) is spanned by the elements e, f , h, so that [h, e] = 2e, [h, f ] = −2f , [e, f ] = h. Consider the element Ω = 1 2h⊗h+e⊗f+f⊗e. Then the map yi,j −→ Ωi,j ∈ A⊗n defines a representation of the Kohno–Drinfeld algebra 4Tn on that A⊗n. The element KZj defined above, corresponds to the truncated (or at critical level) rational Knizhnik– Zamolodchikov operator. Cf. Section 4.2.1, Example 4.49. Proposition 4.85 (T. Kohno, V. Drinfeld). Hilb(4Tn, t) = n−1∏ j=1 (1− jt)−1 = ∑ k≥0 { n+ k − 1 n− 1 } tk, where { n k } stands for the Stirling numbers of the second kind, i.e., the number of ways to partition a set of n things into k nonempty subsets. Remark 4.86. It follows from [6] that Hilb(4Tn, t) is equal to the generating function 1 + ∑ d≥1 v (n) d td for the number v (n) d of Vassiliev invariants of order d for n-strand braids. Therefore, one has the following equality: v (n) d = { n+ d− 1 n− 1 } , i.e., the number of Vassiliev invariants of order d for n-strand braids is equal to the Stirling number of the second kind { n+d−1 n−1 } . On Some Quadratic Algebras 83 We expect that the generating function 1 + ∑ d≥1 v̂ (n) d td for the number v̂ (n) d of Vassiliev invariants of order d for n-strand virtual braids is equal to the Hilbert series Hilb(4NTn, t) of the nonsymmetric Kohno–Drinfeld algebra 4NTn, see Sec- tion 4.3.2. Proposition 4.87 (cf. [7]). The algebra 4NTn, t) is Koszul, and Hilb(4NTn, t) = ( n−1∑ k=0 (k + 1)!N(k, n)(−t)k )−1 , Hilb ( (4NTn)!, t ) = (n− 1)!L (α=1) n−1 ( −t−1 ) tn−1, where N(k, n) := 1 n ( n k )( n k+1 ) denotes the Narayana number, i.e., the number of Dyck n-paths with exactly k peaks, L(α) n (x) = xαex n! dn dxn ( exx n+α ) denotes the generalized Laguerre polynomial. See also Theorem 4.91 below. It is well-known that the quadratic dual 4T ! n of the Kohno–Drinfeld algebra 4Tn is isomorphic to the Orlik–Solomon algebra of type An−1, as well as the algebra 3T anti n . However the algebra 4T 0 n is failed to be Koszul. Examples 4.88. Hilb ( 4T 0 3 , t ) = [2]2[3], Hilb ( 4T 0 4 , t ) = (1, 6, 19, 42, 70, 90, 87, 57, 23, 6, 1), Hilb (( 4T 0 3 )! , t ) (1− t) = (1, 2, 2, 1), Hilb (( 4T 0 4 )! , t ) (1− t)2 = (1, 4, 6, 2,−4,−3), Hilb (( 4T 0 5 )! , t ) (1− t)2 = (1, 8, 26, 40, 24,−3,−6). We expect that Hilb ( (4T 0 n)!, t ) is a rational function with the only pole at t = 1 of order [n/2], cf. Examples 4.77. Remark 4.89. One can show that if n ≥ 4, then Hilb(4T 0 n , t) < Hilb(3T 0 n , t) contrary to the statement of Conjecture 9.6 from [67]. 4.3.2 Nonsymmetric Kohno–Drinfeld algebra 4NTn, and McCool algebras PΣn and PΣ+ n Definition 4.90. The nonsymmetric 4-term relations algebra (or the nonsymmetric Kohno– Drinfeld algebra) 4NTn is an associative algebra (say over Q) with the set of generators yi,j , 1 ≤ i 6= j ≤ n, subject to the following relations 1) yi,j and yk,l are commute if i, j, k, l are all distinct; 2) [yi,j , yi,k + yj,k] = 0 if i, j, k are all distinct. We denote by 4NT+ n the quotient of the algebra 4NTn by the two-sided ideal generated by the elements {yij + yji = 0, 1 ≤ i 6= j ≤ n}. 84 A.N. Kirillov Theorem 4.91. One has Hilb(4NTn, t) = Hilb(CYBn, t), Hilb ( 4NT+ n , t ) = Hilb(6Tn, t) for all n ≥ 2. We expect that the both algebras 4NTn and 4NT+ n are Koszul. Definition 4.92. (1) Define the McCool algebra PΣn to be the quotient of the nonsymmetric Kohno–Drinfeld algebra 4NTn by the two-sided ideal generated by the elements {yikyjk − yjkyik} for all pairwise distinct i, j and k. (2) Define the upper triangular McCool algebra PΣ+ n to be the quotient of the McCool algebra PΣn by the two-sided ideal generated by the elements {yij + yji}, 1 ≤ i 6= j ≤ n. Theorem 4.93. The quadratic duals of the algebras PΣn and PΣ+ n have the following Hilbert polynomials Hilb ( PΣ! n, t ) = (1 + nt)n−1, Hilb (( PΣ+ n )! , t ) = n−1∏ j=1 (1 + jt). Proposition 4.94. (1) The quadratic dual PΣ! n of the algebra PΣn admits the following description. It is gene- rated over Z by the set of pairwise anticommuting elements {yij , 1 ≤ i 6= j ≤ n}, subject to the set of relations (a) y2 ij = 0, yijyji = 0, 1 ≤ i 6= j ≤ n, (b) yikyjk = 0 for all distinct i, j, k, (c) yijyjk + yikyij + ykjyik = 0 for all distinct i, j, k. (2) The quadratic dual (PΣ+ n )! of the algebra PΣ+ n admits the following description. It is generated over Z by the set of pairwise anticommuting elements {zij , 1 ≤ i < j ≤ n}, subject to the set of relations (a) z2 ij = 0 for all i < j, (b) zijzjk = zijzik for all 1 ≤ i < j < k ≤ n. Comments 4.95 (the McCool groups and algebras). The McCool group PΣn is, by definition, the group of pure symmetric automorphisms of the free group Fn consisting of all automorphism that, for a fixed basis {x1, . . . , xn}, send each xi to a conjugate of itself. This group is generated by automorphisms αij , 1 ≤ i 6= j ≤ n, defined by αij(xk) = { xjxix −1 j , k = i, xk, k 6= i. McCool have proved that the relations [αij , αkl] = 1, i, j, k, l are distinct, [αij , αji] = 1, i 6= j, [αij , αikαjk] = 1, i, j, k are distinct. form the set of defining relations for the group PΣn The subgroup of PΣn generated by the αij for 1 ≤ i < j ≤ n is denoted by PΣ+ n and is called by upper triangular McCool group. It is easy to see that the McCool algebras PΣn and PΣ+ n are the “infinitesimal deformations” of the McCool groups PΣn and PΣ+ n respectively. On Some Quadratic Algebras 85 Theorem 4.96. (1) There exists a natural isomorphism H∗(PΣn,Z) ' PΣ! n of the quadratic dual PΣ! n of the McCool algebra PΣn and the cohomology ring H∗(PΣn,Z) of the McCool group PΣn, see [61]. (2) There exists a natural isomorphism H∗(PΣ+ n ,Z) ' (PΣ+ n )! of the quadratic dual (PΣ+ n )! of the upper triangular McCool algebra PΣ+ n and the coho- mology ring H∗(PΣ+ n ,Z) of the upper triangular McCool group PΣ+ n , see [27]. 4.3.3 Algebras 4TTn and 4STn Definition 4.97. (I) Algebra 4TTn is generated over Z by the set of elements {xij , 1 ≤ i 6= j ≤ n}, subject to the set of relations (1) xijxkl = xklxij if all i, j, k, l are distinct, (2) [xij + xjk, xik] = 0, [xji + xkj , xki] = 0 if i < j < k. (II) Algebra 4STn is generated over Z by the set of elements {xij , 1 ≤ i 6= j ≤ n}, subject to the set of relations (1) [xij , xkl] = 0, [xij , xji] = 0 if i, j, k, l are distinct, (2) [xij , xik] = [xik, xjk] = [xjk, xij ], [xji, xki] = [xki, xkj ] = [xkj , xii], (3) [xij , xki] = [xkj , xij ] = [xji, xik] = [xik, xkj ] = [xki, xjk] = [xjk, xji] if i < j < k. Proposition 4.98. One has t ∑ n≥2 Hilb ( (4TTn)!, t )zn n! = exp(−tz) (1− z)2t − 1− tz. Therefore, dim(4TTn)! is equal to the number of permutations of the set [1, . . . , n + 1] having no substring [k, k + 1]; also, for n ≥ 1 equals to the maximal permanent of a nonsingular n× n (0, 1)-matrix, see [131, A000255]47. Moreover, one has Hilb ( (4STn)!, t ) = (1 + t)n(1 + nt)n−2, cf. Conjecture 4.112. We expect that The both algebras 4TTn and 4STn are Koszul. Problem 4.99. Give a combinatorial interpretation of polynomials Hilb((4TTn)!, t) and con- struct a monomial basis in the algebras (4TTn)! and 4STn. 47See also a paper by F. Hivert, J.-C. Novelli and J.-Y. Thibon [57, Section 3.8.4] for yet another combinatorial interpretation of the dimension of the algebra (4TTn)!. 86 A.N. Kirillov 4.4 Subalgebra generated by Jucys–Murphy elements in 4T 0 n Definition 4.100. The Jucys–Murphy elements dj , 2 ≤ j ≤ n, in the quadratic algebra 4Tn are defined as follows dj = ∑ 1≤i<j yi,j , j = 2, . . . , n. It is clear that Jucys–Murphy’s elements dj are the infinitesimal deformation of the elements D1,j ∈ Pn. Theorem 4.101. (1) The Jucys–Murphy elements dj, 2 ≤ j ≤ n, commute pairwise in the algebra 4Tn. (2) In the algebra 4T 0 n the Jucys–Murphy elements dj, 2 ≤ j ≤ n, satisfy the following relations (d2 + · · ·+ dj)d 2j−3 j = 0, 2 ≤ j ≤ n. (3) Subalgebra (over Z) in 4T 0 n generated by the Jucys–Murphy elements d2, . . . , dn has the following Hilbert polynomial n−1∏ j=1 [2j]. (4) There exists an (birational) isomorphism Z[x1, . . . , xn−1]/Jn−1 −→ Z[d2, . . . , dn] defined by dj := n−j∏ i=1 xi, 2 ≤ j ≤ n, where Jn−1 is a (two-sided) ideal generated by ei(x 2 1, . . . , x 2 n−1), 1 ≤ i ≤ n − 1, and ei(x1, . . . , xn−1) stands for the i-th elementary symmetric polynomial in the variables x1, . . . , xn−1. Remark 4.102. (1) It is clearly seen that the commutativity of the Jucys–Murphy elements is equivalent to the validity of the Kohno–Drinfeld relations and the locality relations among the generators {yi,j}1≤i<j≤n. (2) Let’s stress that d2j−2 j 6= 0 in the algebra 4T 0 n for j = 3, . . . , n. For example, d4 3 = y13y23y13y23 + y23y13y23y13 6= 0 since dim(4T 0 3 )4 = 1 and it is generated by the element d4 3. (3) The map ι : yi,j −→ yn+1−j,n+1−i preserves the relations 1) and 2) in the definition of the algebra 4Tn, and therefore defines an involution of the Kohno–Drinfeld algebra. Hence the elements d̂j := n∑ k=j+1 yj,k = ι(dn+1−j), 1 ≤ j ≤ n− 1 also form a pairwise commuting family. Problems 4.103. (a) Compute Hilbert series of the algebra 4T 0 n and its quadratic dual algebra (4T 0 n)!. (b) Describe subalgebra in the algebra 4HTn generated by the Jucys–Murphy elements dj, 2 ≤ j ≤ n. It is well-known that the Kohno–Drinfeld algebra 4Tn is Koszul, and its quadratic dual 4T ! n is isomorphic to the anticommutative quotient 3T 0,anti n of the algebra 3T (−),0 n . On the other hand, if n ≥ 3 the algebra 4T 0 n is not Koszul, and its quadratic dual is isomorphic to the quotient of the ring of polynomials in the set of anticommutative variables {ti,j \| 1 ≤ i < j ≤ n}, where we do not impose conditions t2ij = 0, modulo the ideal generated by Arnold’s relations {ti,jtj,k + ti,k(ti,j − tj,k) = 0} for all pairwise distinct i, j and k. On Some Quadratic Algebras 87 4.5 Nonlocal Kohno–Drinfeld algebra NL4Tn Definition 4.104. Nonlocal Kohno–Drinfeld algebra NL4Tn is an associative algebra over Z with the set of generators {yij , 1 ≤ i < j ≤ n} subject to the set of relations (1) yijykl = yklyij if (i− k)(i− l)(j − k)(j − l) > 0, (2) [ yij , j∑ a=i yak ] = 0 if i < j < k, (3) [ yjk, k∑ a=j yia ] = 0 if i < j < k. It’s not difficult to see that relations (1)–(3) imply the following relations (4) [ xij , j−1∑ a=i+1 (yia + yaj) ] = 0 if i < j. Let’s introduce in the nonlocal Kohno–Drinfeld algebra NL4Tn the Jucys–Murphy elements (JM-elements for short) dj and the dual JM-elements d̂j as follows dj = j−1∑ a=1 yaj , d̂j = n∑ a=n−j+2 yn−j+1,a, j = 2, . . . , n. (4.5) It follows from relations (1) and (2) (resp. (1) and (3)) that the Jucys–Murphy elements d2, . . . , dn (resp. d̂2, . . . , d̂n) form a commutative subalgebra in the algebra NL4Tn. Moreover, it follows from relations (1)–(3) that the element c1 := n∑ j=2 dj = n∑ j=2 d̂j belongs to the center of the algebra NL4Tn. Theorem 4.105. (1) The algebra NL4Tn is Koszul, and Hilb ( (NL4Tn)!, t ) = n−1∑ k=0 Ck ( n+ k − 1 2k ) tk, where Ck = 1 k+1 ( 2k k ) stands for the k-th Catalan number. (2) The quadratic dual (NL4Tn)! of the nonlocal Kohno–Drinfeld algebra NL4Tn is an associa- tive algebra generated by the set of mutually anticommuting elements {tij , 1 ≤ i < j ≤ n} subject to the set of relations • t2ij = 0 if 1 ≤ i < j ≤ n, • Arnold’s relations: tijtjk + tiktij + tjktik = 0 if i < j < k, • disentanglement relations: tiktjl + tiltik + tjltil = 0 if i < j < k < l. Therefore the algebra (NL4Tn)! is the quotient of the Orlik–Solomon algebra OSn by the ideal generated by Disentanglement relations, and dim((NL4Tn+1)!) is equal to the number of Schröder paths, i.e., paths from (0, 0) to (2n, 0) consisting of steps U = (1, 1), D = (1,−1), H = (2, 0) and never going below the x-axis. The Hilbert polynomial Hilb((NL4Tn)!, t) is the generating function of such paths with respect to the number of U ′s, see [131, A088617]. Remark 4.106. Denote byHn(q) “the normalized” Hecke algebra of type An, i.e., an associative algebra generated over Z[q, q−1] by elements T1, . . . , Tn−1 subject to the set of relations 88 A.N. Kirillov (a) TiTj = TjTi if \|i− j\| > 1, TiTjTi = TjTiTj if \|i− j\| = 1, (b) T 2 i = (q − q−1) Ti + 1 for i = 1, . . . , n− 1. If 1 ≤ i < j ≤ n− 1, let’s consider elements T(ij) := TiTi+1 · · ·Tj−1TjTj−1 · · ·Ti+1Ti. Lemma 4.107. The elements {T(ij), 1 ≤ i < j < n − 1} satisfy the defining relations of the non-local Kohno–Drinfeld algebra NL4Tn−1, see Definition 4.104. Therefore the map yij → H(ij) defines a epimorphism ιn : NL4Tn −→ Hn+1(q). Definition 4.108. Denote by NL4Tn the quotient of the non-local Kohno–Drinfeld algebra NL4Tn by the two-sided ideal In generated by the following set of degree three elements: (1) zij := yi,j+1yijyj,j+1 − yj,j+1yijyi,j+1 if 1 ≤ i < j ≤ n, (2) ui := yi,i+1  i−1∑ a=1 i−1∑ b=1, b 6=a yaiyb,i+1 −  i−1∑ a=1 i−1∑ b=1, b 6=a yb,i+1yai  yi,i+1 if 1 ≤ i ≤ n− 1, (3) vi := yi,i+1  n∑ a=i+1 n∑ b=i+1, b 6=a yi+1,ayi,b −  n∑ a=i+1 n∑ b=i+1, b 6=a yi+1,ayi,b  yi,i+1, if 1 ≤ i ≤ n− 1. Proposition 4.109. (1) The ideal Tn belongs to the kernel of the epimorphism ιn : In ⊂ Ker(ιn), (2) Let d2, . . . , dn (resp. d̂2, . . . , d̂n) be the Jucys–Murphy elements (resp. dual JM-elements) in the algebra NL4Tn given by the formula (4.5). Then the all elementary symmetric polynomials ek(d2, . . . , dn) (resp. ek(d̂2, . . . , d̂n)) of deg- ree k, 1 ≤ k < n, in the Jucys–Murphy elements d2, . . . , dn, (resp. in the dual JM-elements d̂2, . . . , d̂n), commute in the algebra NL4Tn with the all elements yi,i+1, i = 1, . . . , n− 1. Therefore, there exists an epimorphism of algebras NL4Tn −→ Hn(q), and images of the ele- ments ek(d2, . . . , dn) (resp. ek(d̂2, . . . , d̂n), 1 ≤ k < n, belongs to the center of the “normalized” Hecke algebra Hn(q), and in fact generate the center of algebra Hn(q). Few comments in order: (A) Let N`4Tn be an associative algebra over Z with the set of generators {yij , 1 ≤ i < j ≤ n} subject to the set of relations (1) yijykl = yklyij if (i− k)(i− l)(j − k)(j − l) > 0, (2) [ yij , j∑ a=i yak ] = 0 if i < j < k. Proposition 4.110. (1) The algebra N`4Tn is Koszul and has the Hilbert series equals to Hilb(N`4Tn, t) = ( n−1∑ k=0 (−1)kN(k, n)tk )−1 , where N(k, n) := 1 n ( n k )( n k+1 ) denotes the Narayana number, i.e., the number of Dyck n- paths with exactly k peaks, see, e.g., [131, A001263]. Therefore, dim(N`4Tn)! = 1 n+1 ( 2n n ) , the n-th Catalan number. On Some Quadratic Algebras 89 (2) Elementary symmetric polynomials ek(d2, . . . , dn) of degree k, 1 ≤ k < n, in the Jucys– Murphy elements d2, . . . , dn, commute in the algebra N`4Tn with the all elements yi,i+1, i = 1, . . . , n− 1. (B) The kernel of the epimorphism NL4Tn −→ Hn(q) contains the elements {yi,i+1yi+1,i+2yi,i+1 − yi+1,i+2yi,i+1yi+1,i+2, i = 1, . . . , n− 2},{ T 2 i,i+1 − ( q − q−1 ) Ti,i+1 − 1 } , as well as the following set of commutators [yij , ek(di, . . . , dj)], 1 ≤ k ≤ j − i+ 1. It is an interesting task to find defining relations among the Jucys–Murphy elements {dj , j = 2, . . . , n} in the algebra NL4Tn or that N`4Tn. We expect that the Jucys–Murphy element dk satisfies the following relation (= minimal polynomial) in the Hecke algebra Hn(q), n ≥ k, k−1∏ a=1 ( dk − q − q2a+1 1− q2 )( dk + q−1 − q−2a−1 1− q−2 ) = 0. 4.5.1 On relations among JM-elements in Hecke algebras Let Hn(q) be the “normalized” Hecke algebra of type An, see Remark 4.106. Let λ ` n be a partition of n. For a box x = (i, j) ∈ λ define cλ(x; q) := q 1− q2(j−i) 1− q2 . It is clear that if q = 1, cq=1(x) is equal to the content c(x) of a box x ∈ λ. Denote by Λ(n) q = Z [ q, q−1 ] [z1, . . . , zn]Sn the space of symmetric polynomials over the ring Z[q, q−1] in variables {z1, . . . , zn}. Definition 4.111. Denote by J (n) q the set of symmetric polynomials f ∈ Λ (n) q such that for any partition λ ` n one has f(cλ(x; q)\|x ∈ λ) = 0. For example, one can check that symmetric polynomial e2 1 − ( q2 + 1 + q−2 ) e2 − 2 ( q − q−1 ) e1 − 3 belongs to the set J (3) q . Finally, denote by J(n) q the ideal in the ring Z[q, q−1][z1, . . . , zn] generated by the set J (n) q . Conjecture 4.112. The algebra over Z[q, q−1] generated by the Jucys–Murphy elements d2, . . . , dn corresponding to the Hecke algebra Hn(q) of type An−1, is isomorphic to the quotient of the algebra Z[q, q−1][z1, . . . , zn] by the ideal J(n) q . It seems an interesting problem to find a minimal set of generators for the ideal J(n) q . 90 A.N. Kirillov Comments 4.113. Denote by JM(n) the algebra over Z generated by the JM-elements d2, . . . , dn, deg(di) = 1, ∀ i, corresponding to the symmetric group Sn. In this case one can check Conjecture 4.112 for n < 8, and compute the Hilbert polynomial(s) of the associated graded algebra(s) gr(JM(n)). For example48 Hilb(gr(JM(2), t) = (1, 1), Hilb(gr(JM(3), t) = (1, 2, 1), Hilb(gr(JM(4), t) = (1, 3, 4, 2), Hilb(gr(JM(5), t) = (1, 4, 8, 9, 4), Hilb(gr(JM(6), t) = (1, 5, 13, 21, 21, 12, 3), Hilb(gr(JM(7), t) = (1, 6, 19, 40, 59, 60, 37, 10). It seems an interesting task to find a combinatorial interpretation of the polynomials Hilb(gr(JM(n)), t) in terms of standard Young tableaux of size n. Let {χλ, λ ` n} be the characters of the irreducible representations of the symmetric group Sn, which form a basis of the center Zn of the group ring Z[Sn]. The famous result by A. Jucys [62] states that for any symmetric polynomial f(z1, . . . , zn) the character expansion of f(d2, . . . , dn, 0) ∈ Zn is f(d2, . . . , dn, 0) = ∑ λ`n f(Cλ) Hλ χλ, where Hλ = ∏ x∈λ hx denotes the product of all hook-lengths of λ, and Cλ := {c(x)}x∈λ denotes the set of contents of all boxes of λ. Recall that the Jucys–Murphy elements { dHj } 2≤j≤n in the (normalized) Hecke algebra Hn(q) are defined as follows: dHj := ∑ i<j T(ij), where T(ij) := Ti · · ·Tj−1TjTj−1 · · ·Ti. Finally denote by Hλ(q) and C (q) λ the hook polynomial and the set {cλx; q)}x ∈ λ. Then for any symmetric polynomial f(z1, . . . , zn) one has f ( dH2 , . . . , d H n , 0 ) = ∑ λ`n f(C (q) λ ) Hλ(q) χλq , where χλq denotes the q-character of the algebra Hn(q). Therefore, if f ∈ J (n) q , then f ( dH2 , . . . , d H n , 0 ) = 0. It is an open problem to prove/disprove that if f ( dH2 , . . . , d H n , 0 ) = 0, then f ( C (q) λ ) = 0 for all partitions of size n (even in the case q = 1). 4.6 Extended nil-three term relations algebra and DAHA, cf. [24] Let A := {q, t, a, b, c, h, e, f, . . .} be a set of parameters. Definition 4.114. Extended nil-three term relations algebra 3Tn is an associative algebra over Z[q±1, t±1, a, b, c, h, e, . . .] with the set of generators {ui,j , 1 ≤ i 6= j ≤ n, xi, 1 ≤ i ≤ n, π} subject to the set of relations (0) ui,j + uj,i = 0, u2 i,j = 0, (1) xixj = xjxi, ui,juk,l = uk,lui,j if i, j, k, l are distinct, (2) xiukl = uk,lxi if i 6= k, l, 48I would like to thank Dr. S. Tsuchioka for computation the Hilbert polynomials Hilb(JM(n), t), as well as the sets of defining relations among the Jucys–Murphy elements in the symmetric group Sn for n ≤ 7. On Some Quadratic Algebras 91 (3) xiui,j = ui,jxj + 1, xjui,j = ui,jxi − 1, (4) ui,juj,k + uk,iui,j + uj,kuk,i = 0 if i, j, k are distinct, (5) πxi = xi+1π if 1 ≤ i < n, πxn = t−1x1π, (6) πuij = ui+1,j+1 if 1 ≤ i < j < n, πjun−j+1,n = tu1,jπ j , 2 ≤ j ≤ n. Note that the algebra 3Tn contains also the set of elements {πaujn, 1 ≤ a ≤ n− j}. Definition 4.115 (cf. [87]). Let 1 ≤ i < j ≤ n, define Ti,j = a+ (bxi + cxj + h+ exixj)ui,j . Lemma 4.116. (1) T 2 i,j = (2a+ b− c)Ti,j − a(a+ b− c) if a = 0, then T 2 ij = (b− c)Tij. (2) Coxeter relations Ti,jTj,kTi,j = Tj,kTi,jTj,k, are valid, if and only if the following relation holds (a+ b)(a− c) + he = 0. (4.6) (3) Yang–Baxter relations Ti,jTi,kTj,k = Tj,kTi,kTi,j are valid if and only if b = c = e = 0, i.e., Tij = a+ duij. (4) T 2 ij = 1 if and only if a = ±1, c = b± 2, he = (b± 1)2. (5) Assume that parameters a, b, c, h, e satisfy the conditions (4.6) and that bc + 1 = he. Then TijxiTij = xj + (h+ (a+ b)(xi + xj) + exixj)Tij . (6) Quantum Yang–Baxterization. Assume that parameters a, b, c, h, e satisfy the condi- tions (4.6) and that β := 2a + b − c 6= 0. Then (cf. [59, 85] and the literature quoted therein) the elements Rij(u, v) := 1 + λ−µ βµ Tij satisfy the twisted quantum Yang–Baxter relations Rij(λi, µj)Rjk(λi, νk)Rij(µj , νk) = Rjk(µj , νk)Rij(λi, νk)Rjk(λi, µj), i < j < k, where {λi, µi, νi}1≤i≤n are parameters. Corollary 4.117. If (a+ b)(a− c) + he = 0, then for any permutation w ∈ Sn the element Tw := Ti1 · · ·Til ∈ 3Tn, where w = si1 · · · sil is any reduced decomposition of w, is well-defined. Example 4.118. Each of the set of elements s (h) i = 1 + (xi+1 − xi + h)ui,i+1 and t (h) i = −1 + (xi − xi+1 + h(1 + xi)(1 + xi+1)uij , i = 1, . . . , n− 1, by itself generate the symmetric group Sn. 92 A.N. Kirillov Comments 4.119. Let A = (a, b, c, h, e) be a sequence of integers satisfying the conditions (4.6). Denote by ∂Ai the divided difference operator ∂Ai = (a+ (bxi + cxi+1 + h+ exixi+1)∂i, i = 1, . . . , n− 1. It follows from Lemma 4.107 that the operators {∂Ai }1≤i≤n satisfy the Coxeter relations ∂Ai ∂ A i+1∂ A i = ∂Ai+1∂ A i ∂ A i+1, i = 1, . . . , n− 1. Definition 4.120. (1) Let w ∈ Sn be a permutation. Define the generalized Schubert polynomial corresponding to permutation w as follows SA w(Xn) = ∂Aw−1w0 xδn , where xδn := xn−1 1 xn−2 2 · · ·xn−1, and w0 denotes the longest element in the symmetric group Sn. (2) Let α be a composition with at most n parts, denote by wα ∈ Sn the permutation such that wα(α) = α, where α denotes a unique partition corresponding to composition α. Proposition 4.121 ([71]). Let w ∈ Sn be a permutation. • If A = (0, 0, 0, 1, 0), then SA w(Xn) is equal to the Schubert polynomial Sw(Xn). • If A = (−β, β, 0, 1, 0), then SA w(Xn) is equal to the β-Grothendieck polynomial G (β) w (Xn) introduced in [42]. • If A = (0, 1, 0, 1, 0) then SA w(Xn) is equal to the dual Grothendieck polynomial [71, 84]. • If A = (−1, 2, 0, 1, 1), then SA w(Xn) is equal to the Di Francesco–Zinn-Justin polynomials and studied in [32, 33, 34] and [71]. In all cases listed above the polynomials SA w(Xn) have non-negative integer coefficients. • If A = (1,−1, 1,−h, 0), then SA w(Xn) is equal to the h-Schubert polynomials introduced in [71]. Define the generalized key or Demazure polynomial corresponding to a composition α as follows KA α (Xn) = ∂wαx α. • If A = (1, 0, 1, 0, 0), then KA α (Xn) is equal to key (or Demazure) polynomial corresponding to α. • If A = (0, 0, 1, 0, 0), then KA α (Xn) is equal to the reduced key polynomial introduced in [71]. • If A = (1, 0, 1, 0, β), then KA α (Xn) is equal to the key Grothendieck polynomial KGα(Xn) introduced in [71]. • If A = (0, 0, 1, 0, β), then KA α (Xn) is equal to the reduced key Grothendieck polynomial [71]. In all cases listed above the polynomials SA w(Xn) have non-negative integer coefficients. On Some Quadratic Algebras 93 Exercises 4.122. (1) Let b, c, h, e be a collection of integers, define elements Pij := fijuij ∈ 3T, where fij := bxi + cxj + h+ exixj . Show that • P 2 ij = (b− c)Pij , • PijPjkPij = fijfikfjkuijujkuij + (bc− eh)Pij , PjkPijPjk = fijfikfjkuijujkuij − (bc− eh)Pjk. (2) Assume that a = q, b = −q, c = q−1, h = e = 0, and introduce elements eij := ( qxi − q−1xj ) uij , 1 ≤ i < j < k ≤ n. (a) Show that if i, j, k are distinct, then eijejkeij = eij + ( qxi − q−1xj )( qxi − q−1xk )( qxj − q−1xk ) uijujkuij , e2 ij = ( q + q−1 ) eij . (b) Assume additionally that uijujkuij = 0, if i, j, k are distinct. Show that the elements {ei := ei,i+1, i = 1, . . . , n− 1}, generate a subalgebra in 3Ln which is isomorphic to the Temperley–Lieb algebra TLn(q + q−1). (3) Let us set Ti := Ti,i+1, i = 1, . . . , n− 1, and define T0 := πTn−1π −1. Show that if (a+ b)(a− c) + eh = 0, then T1T0T1 = T1T0T1, Tn−1T0Tn−1 = T0Tn−1T0, Recall that T 2 i = (2a+ b− c)Ti − a(a+ b− c), 0 ≤ i ≤ n− 1. In what follows we take a = q, b = −q, c = q−1, h = e = 0. Therefore, T 2 i,j = (q−q−1)Ti,j +1. We denote by Hn(q) a subalgebra in 3Tn generated by the elements Ti := Ti,i+1, i = 1, . . . , n−1. Remark 4.123. Let us stress on a difference between elements Tij as a part of generators of the algebra 3Tn, and the elements T(ij) := Ti · · ·Tj−1TjTj−1 · · ·Ti ∈ Hn(q). Whereas one has [Tij , Tkl] = 0 if i, j, k, l are distinct, the relation [T(ij), T(kl)] = 0 in the algebra Hn(q) holds (for general q and i ≤ k) if and only if either one has i < j < k < l or i < k < l < j. Lemma 4.124. (1) TijTkl = TklTij if i, j, k, l are distinct, (2) Ti,jxiTi,j = xj if 1 ≤ i < j ≤ n, (3) πTi,j = Ti+1,j+1 if 1 ≤ i < j < n, πjTn−j+1,n = T1,jπ j. Definition 4.125. Let 1 ≤ i < j ≤ n, set Yi,j = T−1 i−1,j−1T −1 i−2,j−2 · · ·T −1 1,j−i+1π j−i Tn−j+i,n · · ·Ti+1,j+1Ti,j , 1 ≤ i < j ≤ n, and Yn = T−1 n−1,n · · ·T −1 1,2 π. 94 A.N. Kirillov For example, Y1,j = πj−1Tn−j+1,n · · ·T1,j , j ≥ 2, Y2,j = T−1 1,j−1π j−2Tn−j+2,n · · ·T2,j , and so on, Yj−1,j = T−1 j−2,j−1 · · ·T −1 1,2 πTn−1,n · · ·Tj−1,j . Proposition 4.126. (1) xjxjTij = Tijxixj, (2) Yi,j = Ti,jYi+1,j+1Ti,j if 1 ≤ i < j < n, (3) Yi,jYi+k,j+k = Yi+k,j+kYi,j if 1 ≤ i < j ≤ n− k, (4) one has xi−1Y −1 i,j = Y −1 i,j xi−1T 2 i−1,j−1, 2 ≤ i < j ≤ n, (5) Yi,jx1x2 · · ·xn = tx1x2 · · ·xnYi,j, (6) xiY1Y2 · · ·Yn = t−1Y1Y2 · · ·Ynxi, where we set Yi := Yi,i+1, 1 ≤ i < j < n. Conjecture 4.127. Subalgebra of 3Tn generated by the elements {Ti := Ti,i+1, 1 ≤ i < n, Y1, . . . , Yn, and x1, . . . , xn}, is isomorphic to the double affine Hecke algebra DAHAq,t(n). Note that the algebra 3Tn contains also two additional commutative subalgebras generated by additive { θi = ∑ j 6=i uij } 1≤i≤n and multiplicative { Θi = i−1∏ a=1 (1− uai) n∏ a=i+1 (1 + uia) } 1≤i≤n Dunkl elements correspondingly. Finally we introduce (cf. [24]) a (projective) representation of the modular group SL(2,Z) on the extended affine Hecke algebra Ĥn over the ring Z[q±1, t±1] generated by elements {T1, . . . , Tn−1}, π, and {x1, . . . , xn}. It is well-known that the group SL(2,Z) can be generated by two matrices τ+ = ( 1 1 0 1 ) , τ− = ( 1 0 1 1 ) , which satisfy the following relations τ+τ −1 − τ+ = τ−1 − τ+τ −1 − , ( τ+τ −1 − τ+ )6 = I2×2. Let us introduce operators τ+ and τ− acting on the extended affine algebra Ĥn. Namely, τ+(π) = x1π, τ+(Ti) = Ti, τ+(xi) = xi, ∀ i, τ−(π) = π, τ−(Ti) = Ti, τ−(xi) = ( 1∏ a=i−1 Ta ) π ( i∏ a=n Ta ) xi. On Some Quadratic Algebras 95 Lemma 4.128. τ+(Yi) = ( 1∏ a=i−1 T−1 a )( i−1∏ a=1 T−1 a ) xiYi, τ−(xi) = ( 1∏ a=i−1 Ta )( i−1∏ a=1 Ta ) Yixi,( τ+τ −1 − τ+ ) (xi) = Y −1 i = ( τ−1 − τ+τ −1 − ) (xi),( τ+τ −1 − τ+ ) (Yi) = txi ( 1∏ a=i−1 Ta ) (T1 · · ·Tn−1) ( i∏ a=n−1 Ta ) , i = 1, . . . , n. In the last formula we set Tn = 1 for convenience. 4.7 Braid, affine braid and virtual braid groups The main objective of this section is to describe the distinguish abelian subgroup in the braid group Bn, see Proposition 4.132 ( 2(0) ) , and that in the Yang–Baxter groups ŶBn and YBn, see Proposition 4.132 ( 5(0) ) and ( 6(0) ) correspondingly. As far as I’m aware, these constructions go back to E. Artin in the case of braid groups, and to C.N. Yang in the case of Yang–Baxter group, and nowadays are widely use in the representation theory of Hecke’s type algebras and that of integrable systems. In a few words, by choosing a suitable representation (finite-dimensional or birational) of either Bn or YBn, or ŶBn, one gives rise to a family of mutually commuting operators acting in the space of a representation selected. In the case of braid groups one comes to Jucys–Murphy’s type operators/elements, and in the case of Yang–Baxter groups one comes to Dunkl’s type operators/elements. See, e.g., [59, 60], where it was used the so-called R-matrix representation of the affine braid group of type C (1) n to construct the (two boundary) quantum Knizhnik–Zamolodchikov connections with values in the affine Birman–Murakami– Wenzl algebras. To start with, let n ≥ 2 be an integer. Definition 4.129. • Denote by Sn the symmetric group on n letters, and by si the simple transposition (i, i+1) for 1 ≤ i ≤ n − 1. The well-known Moore–Coxeter presentation of the symmetric group has the form 〈s1, . . . , sn−1 \| s2 i = 1, sisi+1si = si+1sisi+1, sisj = sjsi if \|i− j\| ≥ 2〉. Transpositions sij := sisi+1 · · · sj−2sj−1sj−2 · · · si+1si, 1 ≤ i < j < j ≤ n, satisfy the following set of (defining) relations s2 ij = 1, sijskl = sklsij if {i, j} ∩ {k, l} = ∅, sijsik = sjksij = siksjk, siksij = sijsjk = sjksik, i < j < k. • The Artin braid group on n strands Bn is defined by generators σ1, . . . , σn−1 and relations σiσi+1σi = σi+1σiσi+1, 1 ≤ i ≤ n− 2, σiσj = σjσi if \|i− j\| ≥ 2. (4.7) • The monoid of positive braids on n strands B+ n is a monoid generated by the elements σ1, . . . , σn−1 subject to the set of relations (4.7). 96 A.N. Kirillov • A new representation of the braid group [15]. The Birman–Ko–Lee representation of the braid group Bn has the set of generators {σi,j \| 1 ≤ i < j ≤ n} subject to the Birman–Ko– Lee (defining) relations σi,jσk,l = σk,lσi,j if (j − l)(j − k)(i− l)(i− k) > 0, σi,jσi,k = σj,kσi,j = σi,kσj,k if 1 ≤ i < j < k ≤ n. One can take σi,j := (σj−1 · · ·σi+1)σi(σ −1 i+1 · · ·σ −1 j−1), see [14] for details. It would be well to note that as a corollary of the Birman–Ko–Lee relations one can deduce the 2D Coxeter relations among the Birman–Ko–Lee generators σi,jσj,kσi,j = σj,kσi,jσj,k, 1 ≤ i < j < k ≤ n. • The Birman–Ko–Lee monoid BKLn is a monoid generated by the elements σi,j , 1 ≤ i < j ≤ n, subject to the Birman–Ko–Lee relations. We denote by BKL(n) and called it as the Birman–Ko–Lee algebra, the group algebra Q[BKLn] of the Birman–Ko–Lee monoid. The Hilbert series of the Birman–Ko–Lee algebra BKL(n) will be computed in Section 4.7.3, Theorem 4.134. • The pure braid group PBn is defined to be the kernel of the natural (non-split) projection p : Bn −→ Sn given by p(σi) = si. It is well-known that the pure braid group PBn is generated by the elements gi,j := σ2 i,j = σj−1σj−2 · · ·σi+1σ 2 i σ −1 i+1 · · ·σ −1 j−2σ −1 j−1 for 1 ≤ i < j ≤ n, subject to the following defining relations gi,jgk,l = gk,lgi,j if (i− k)(i− l)(j − k)(j − l) > 0, gi,jgi,kgj,k = gi,kgj,kgi,j = gj,kgi,jgi,k if 1 ≤ i < j < k ≤ n, gi,kgi,lgj,lgk,l = gi,lgj,lgk,lgi,k if 1 ≤ i < j < k < l ≤ n. Comments 4.130. It is easy to see that the defining relations for the pure braid group PBn listed above are equivalent to the following list of defining relations g−1 i,j gk,lgi,j =  gk,l if (i− k)(i− l)(j − k)(j − l) > 0, gi,lgk,lg −1 i,l if i < k = j < l, gi,lgj,lgk,lg −1 i,l g −1 j,l if i = k < j < l, gi,lgj,lg −1 i,l g −1 j,l gk,lgj,lgi,lg −1 j,l g −1 i,l if i < k < j < l, commonly used in the literature to describe the defining relations among the generators {gij} of the pure braid group Pn, see, e.g., [14]. • The affine Artin braid group Baff n , cf. [112], is an extension of the Artin braid group on n strands Bn by the element τ subject to the set of crossing relations σ1τσ1τ = τσ1τσ1, σiτ = τσi for 2 ≤ i ≤ n− 1. • The virtual braid group VBn is a group generated by σ1, . . . , σn−1 and s1, . . . , sn−1 subject to the relations: (1) braid relations σ1, . . . , σn−1 generate the Artin braid group Bn; (2) Moore–Coxeter relations s1, . . . , sn−1 generate the symmetric group Sn; (3) crossing relations σisj = sjσi if \|i− j\| ≥ 2, sisi+1σi = σi+1sisi+1 if 1 ≤ i ≤ n− 2. • The virtual pure braid group VPn is defined to be the kernel of the natural map η : VBn −→ Sn, η(σi) = η(si) = si, i = 1, . . . , n− 1. On Some Quadratic Algebras 97 4.7.1 Yang–Baxter groups Definition 4.131. • The quasitriangular Yang–Baxter group ŶBn, cf. [7], is a group generated by the set of elements {Qi,j , 1 ≤ i 6= j ≤ n}, subject to the set of defining relations (1) [Qi,j , Qk,l] = 0 if i, j, k and l are all distinct, (2) Yang–Baxter relations Qi,jQi,kQj,k = Qj,kQi,kQi,j if i, j, k are distinct. According to [5, Theorem 1], the quasitriangular Yang–Baxter group ŶBn is isomorphic to the virtual pure braid group VPn. • The Yang–Baxter monoid ỸBn is a monoid generated by the elements Qi,j , 1 ≤ i 6= j ≤ n. Important particular case corresponds to the case when Qi,jQj,i = 1 for all 1 ≤ i 6= j ≤ n. • The Yang–Baxter group YBn is defined by the set of generators Ri,j , 1 ≤ i < j ≤ n, subject to the set of defining relations (1) Ri,jRk,l = Rk,lRi,j if i, j, k and l are pairwise distinct, (2) Ri,jRi,kRj,k = Rj,kRi,kRi,j if 1 ≤ i < j < k ≤ n. 4.7.2 Some properties of braid and Yang–Baxter groups For the sake of convenience and future references, below we state some basic properties of the groups Pn, YBn and Baff n . Proposition 4.132. Let Fm denotes the free group with m generators.( 10 ) The elements g1,n, g2,n, . . . , gn−1,n generate a free normal subgroup Fn−1 in Pn, and Pn = Pn−1 n 〈g1,n, g2,n, . . . , gn−1,n〉. Hence Pn is an iterated extension of free groups.( 20 ) Let us consider the following elements in the group Baff n : γ1 = τ, γi = 1∏ j=i−1 σjτ i−1∏ j=1 σj , 2 ≤ i ≤ n. Then (a) commutativity, γiγj = γjγi for all 1 ≤ i, j ≤ n; (b) the elements γ1, . . . , γn generate a free abelian subgroup of rang n in Baff n .49( 30 ) Let us introduce elements Di,j := σj−1σj−2 · · ·σi+1 σ 2 i σi+1 · · ·σj−2σj−1 = ∏ i≤a<j ga,j ∈ Pn, Fi,j := σn−jσn−j+1 · · ·σn−i−1σ 2 n−iσn−i−1 · · ·σn−j+1σn−j = i+1∏ a=j gn−a,n−i ∈ Pn, where 1 ≤ i < j ≤ n. For example, Di,i+1 = σ2 i , Di,i+2 = σi+1σ 2 i σi+1, Fi,i+1 = σ2 n−i, Fi,i+2 = σn−i−1σ 2 n−iσn−i−1 and etc. Then 49We refer the reader to [112] for more details about affine braid groups. Here we only remark that the type A affine Weyl groups Ŝn, the Hecke algebras Hn,q, the affine Hecke algebras Ĥn,q, the Ariki–Koike, or cyclotomic Hecke, algebras Hr,1,n, the affine and cyclotomic Birman–Murakami–Wenzl algebras Zr,1,n, all can be obtained as certain quotients of the group algebra CBaff n of the affine braid group. 98 A.N. Kirillov • For each j = 3, . . . , n, the element D1,j commutes with σ1, . . . , σj−2. • The elements D1,2, D1,3, . . . , D1,n (resp. F1,2, F1,3, . . . , F1,n) generate a free abelian subgroup in Pn. • If n ≥ 3, the element∏ 2≤j≤n D1,j = ∏ 2≤j≤n F1,j = (σ1 · · ·σn−1)n generates the center of the braid group Bn and that of the pure braid group Pn. • Di,jDi,j+1Dj,j+1 = Dj,j+1Di,j+1Di,j if i < j. • Consider the elements s := σ1σ2σ1, t := σ1σ2 in the braid group B3. Then s2 = t3 and the element c := s2 generates the center of the group B3. Moreover, B3/〈c〉 ∼= PSL2(Z), B3/〈c2〉 ∼= SL2(Z).( 40 ) Let us introduce the following elements in the quasitriangular Yang–Baxter group ŶBn: Ci,j =  i∏ a=j−1 Qa,j (j−1∏ b=i Qj,b ) , fi,j =  i+1∏ a=j−1 Qa,j Qi,jQj,i ( j−1∏ b=i+1 Qb,j )−1 . Then • The elements C1,2, C1,3, . . . , C1,n generate a free abelian subgroup in ŶBn. • The elements fi,j, 1 ≤ i < j ≤ n, generate a subgroup in ŶBn, which is isomorphic to the pure braid group Pn.50( 50 ) Assume that the following additional relations in ŶBn are satisfied Qi,jQj,i = Qj,iQi,j , Qk,lQi,jQj,i = Qj,iQi,jQk,l if i 6= j and k 6= l. In other words, the elements Qi,j and Qj,i commute, and the elements Qi,jQj,i = Qj,iQi,j are central. Under these assumptions, we have that the elements Θi := 1∏ j=i−1 Qj,i i+1∏ j=n Qi,j , Θ̄i := n∏ j=i+1 Qj,i i−1∏ j=1 Qi,j , 1 ≤ i ≤ n, 50It is enough to check that the elements {fi,j , 1 ≤ i < j ≤ n} satisfy the defining relations for the pure braid group Pn only in the case n = 4. Let us prove that f1,4f2,4f3,4f1,3 = f1,3f1,4f2,4f3,4. Other relations are simple and can be checked in a similar fashion. Let l.h.s. = f1,4f2,4f3,4f1,3 = Q34Q24Q14Q41Q42Q43Q23Q13Q31Q −1 23 , r.h.s. = f1,3f1,4f2,4f3,4 = Q23Q13Q31Q −1 23 Q34Q24Q14Q41Q42Q43. Now we are going to apply the Yang–Baxter relations Q−1 23 Q34Q24 = Q24Q34Q −1 23 , Q−1 23 Q42Q43 = Q43Q42Q −1 23 , Q31Q34Q14 = Q14Q34Q31. Therefore, r.h.s. = Q23Q13Q31Q −1 23 Q34Q24Q14Q41Q42Q −1 23 = Q23Q24Q34Q14Q13Q31Q41Q43Q42Q −1 23 = Q34Q24Q14Q23Q13Q43Q41Q42Q31Q −1 23 = Q34Q24Q14Q41Q23Q43Q42Q13Q31Q −1 23 = Q34Q24Q14Q41Q42Q43Q23Q13Q31Q −1 23 = l.h.s. On Some Quadratic Algebras 99 satisfy the following relations [Θi,Θj ] = 0 = [Θ̄i, Θ̄j ], ΘiΘ̄i = ∏ j 6=i Qi,jQj,i = Θ̄iΘi, n∏ i=1 Θi = ∏ 1≤i 6=j≤n Qi,jQj,i = n∏ i=1 Θ̄i. ( 60 ) In the special case Qi,jQj,i = 1 for all i 6= j, the following statement holds: the elements Θj = 1∏ a=j−1 R−1 a,j j+1∏ b=n Rj,b, 1 ≤ j ≤ n− 1, generate a subgroup in the Yang–Baxter group YBn, which is isomorphic to the free abelian group of rang n− 1. 4.7.3 Artin and Birman–Ko–Lee monoids Let (W,S) be a finite Coxeter group, B(W ) and B+(W ) be the corresponding braid group and monoid of positive braids. Denote by PW (s, t) = ∑ i≥0, j≥0 BQ[B+(W )](i, j)s itj the Poincaré polynomial of the group algebra over Q of the monoid B+(W ). Conjecture 4.133. PW (s, 1) = (s+ 1)\|S\|. It is known [30, 125] that the Hilbert series of the group algebra of the monoid B+(W ) is a rational function of the form 1 P (t) for a some polynomial P (t) := PW (t) ∈ Z[t]. Theorem 4.134. (1) Some Betti numbers of the group algebra over Q of the monoid B+(An−1): BQ[B+(An−1)](k, k) = ( n− k k ) , BQ[B+(An−1)] ( k, ( k + 1 2 )) = n− k, 1 ≤ k ≤ n− 1, BQ[B+(An−1)](k, k + 1) = (k − 1) ( n− k k − 1 ) , BQ[B+(An−1)](k, k + 2) = ( k − 2 2 )( n− k k − 2 ) , BQ[B+(An−1)](k, k + 3) = (k − 2) ( n− k k − 2 ) + max(3k − 17, 0) ( n− k k − 3 ) if k ≥ 3. (2) The Birman–Ko–Lee algebra BKL(n) is Koszul, and the Hilbert polynomial of its quadratic dual is equal to Hilb ( BKL(n)!, t ) = n−1∑ k=0 1 k + 1 ( n− 1 k )( n+ k − 1 k ) tk. Conjecture 4.135 (type An−1 case). Let I ⊂ [1, n − 1] be a subset of vertices in the Dynkin diagram of type An−1, and RI denotes the root system generated by the positive roots {αij = εi − εj , (i, j) ∈ I × I}. Assume that RI ∼= An1 ∐ · · · ∐ Ank , n1 + · · ·+ nk = n− 1 100 A.N. Kirillov stands for the decomposition of the root system RI into the disjoint union of irreducible root subsystems of type A. The numbers n1, . . . , nk are defined uniquely up to a permutation. Let us set n(I) = k∑ a=1 ( na 2 ) . Then PAn−1(s, t) = ∑ I s\|I\|tn(I), where the sum runs over the all subsets of vertices I in the Dynkin diagram of type An−1, including the empty set, and \|I\| denotes the cardinality of the set I. Comments 4.136. (A) The Hilbert polynomial of the Birman–Ko–Li algebra BKL(n) has been computed also by M. Albenque and P. Nadeau, see [2]. (B) Let’s consider the truncated theta function θ+(z, t) = ∑ n≥0 tn(n+1)/2zn. Then ∑ n≥1 PAn−1(s, t)zn−1 = θ+(t, sz)/(1− z(θ+(t, sz))). (C) It is well known that the number T (n, k) = 1 k + 1 ( n k ) ( n+ k k ) counts the number of Schröder paths (i.e., consisting of steps (1, 1), (1,−1) and (2, 0) and never going below x-axis) from (0, 0) to (2n, 0), having exactly k (1, 1) steps. In particular, dim(BKL(n)!) = Sch(n), is the n-th (large) Schröder number, see [131, A006318]. It is a classical result that n∑ k=0 T (n, k)xk(1− x)n−k = n−1∑ k=0 N(n, k)xk, where N(n, k) := 1 n ( n k )( n k+1 ) denotes the Narayana number. Some explicit combinatorial interpretations of the values of the above polynomials for x = 0, 1, 2, 4 can be found in [131, A088617]. Note that Hilb(BKL!, t) = (1 + t)Assn−2(t), where Assn(t) := n∑ k=0 1 k + 1 ( n k )( n+ k k ) tk denotes the f -vector polynomial corresponding to the associahedron of type An. (D) The polynomials F (n, t) := ∑ k≥0 BQ[B+(An−1)](k, k)tk appear to be equal to the so-called Fibonacci polynomials, see, e.g., [131, A011973]. It is well-known that∑ n≥0 F (n, t)yn = 1 + ty 1− y − ty2 . Moreover, the coefficient BQ[B+(An−1)](k, k) is equal to the number of compositions of n+2 into k + 1 parts, all ≥ 2, see [131, A011973]. On Some Quadratic Algebras 101 (E) Monoid of positive pure braids. The monoid of positive pure braids PB+ n (of the type An−1) is the monoid generated by the set {gi,j , 1 ≤ i < j ≤ n} of the Artin generators of the pure braid group PBn. Conjecture 4.137. The following list of relations is the defining set of relations in the monoid PBn: (a) [gi,j , gk,l] = 0, [gi,l, gj,k] = 0, if 1 ≤ i < j < k < l ≤ n, (b) [ gj1+m,jk−1+m, k−1∏ a=1 gja+m,jk+m ] = 0, for all sequences of integers 1 ≤ j1 < j2 < · · · < jk ≤ n of the length k ≥ 4 and m = 0, . . . , n−1. Here we assume that gi,j = gj,i for all i 6= j, and for any non-negative integer a we denote by a a unique integer 1 ≤ a ≤ n such that a ≡ a (modn+ 1). It is worth noting that the defining relations in the pure braid group Pn are that listed in (a) and the part of that listed in (b) corresponding to k = 3, m = 0 and 1, and that for k = 4, m = 0. 5 Combinatorics of associative Yang–Baxter algebras Let α and β be parameters. Definition 5.1 ([66, 70, 72], cf. [1, 115]). (1) The associative quasi-classical Yang–Baxter algebra of weight (α, β), denoted by ÂCYBn(α, β), is an associative algebra, over the ring of polynomials Z[α, β], generated by the set of elements {xij , 1 ≤ i < j ≤ n}, subject to the set of relations (a) xijxkl = xklxij if {i, j} ∩ {k, l} = ∅, (b) xijxjk = xikxij + xjkxik + βxik + α if 1 ≤ 1 < i < j ≤ n. (2) Define associative quasi-classical Yang–Baxter algebra of weight β, denoted by ÂCYBn(β), to be ÂCYBn(0, β). Comments 5.2. The algebra 3Tn(β), see Definition 3.1, is the quotient of the algebra ÂCYBn(−β), by the “dual relations” xjkxij − xijxik − xikxjk + βxik = 0, i < j < k. The (truncated) Dunkl elements θi = ∑ j 6=i xij , i = 1, . . . , n, do not commute in the algebra ÂCYBn(β). However a certain version of noncommutative elementary polynomial of degree k ≥ 1, still is equal to zero after the substitution of Dunkl elements instead of variables [72]. We state here the corresponding result only “in classical case”, i.e., if β = 0 and qij = 0 for all i, j. Lemma 5.3 ([72]). Define noncommutative elementary polynomial Lk(x1, . . . , xn) as follows Lk(x1, . . . , xn) = ∑ I=(i1<i2<···<ik)⊂[1,n] xi1xi2 · · ·xik . Then Lk(θ1, θ2, . . . , θn) = 0. Moreover, if 1 ≤ k ≤ m ≤ n, then one can show that the value of the noncommutative polynomial Lk(θ (n) 1 , . . . , θ (n) m ) in the algebra ÂCYBn(β) is given by the Pieri formula, see [45, 117]. 102 A.N. Kirillov 5.1 Combinatorics of Coxeter element Consider the “Coxeter element” w ∈ ÂCYBn(α, β) which is equal to the ordered product of “simple generators”: w := wn = n−1∏ a=1 xa,a+1. Let us bring the element w to the reduced form in the algebra ÂCYBn(α, β), that is, let us consecutively apply the defining relations (a) and (b) to the element w in any order until unable to do so. Denote the resulting (noncommutative) polynomial by Pn(xij ;α, β). In principal, the polynomial itself can depend on the order in which the relations (a) and (b) are applied. We set Pn(xij ;β) := Pn(xij ; 0, β). Proposition 5.4 (cf. [133, Exercise 6.C5(c)], [99, 100]). (1) Apart from applying the relation (a) (commutativity), the polynomial Pn(xij ;β) does not depend on the order in which relations (a) and (b) have been applied, and can be written in a unique way as a linear combination: Pn(xij ;β) = n−1∑ s=1 βn−s−1 ∑ {ia} s∏ a=1 xia,ja , where the second summation runs over all sequences of integers {ia}sa=1 such that n− 1 ≥ i1 ≥ i2 ≥ · · · ≥ is = 1, and ia ≤ n − a for a = 1, . . . , s − 1; moreover, the corresponding sequence {ja}n−1 a=1 can be defined uniquely by that {ia}n−1 a=1 . • It is clear that the polynomial P (xij ;β) also can be written in a unique way as a linear combination of monomials s∏ a=1 xia,ja such that j1 ≥ j2 · · · ≥ js. (2) Let us set deg(xij) = 1, deg(β) = 0. Denote by Tn(k, r) the number of degree k monomials in the polynomial P (xij ;β) which contain exactly r factors of the form x∗,n. (Note that 1 ≤ r ≤ k ≤ n− 1.) Then Tn(k, r) = r k ( n+ k − r − 2 n− 2 )( n− 2 k − 1 ) . In other words, Pn(t, β) = ∑ 1≤r≤k<n Tn(k, r)trβn−1−k, where Pn(t, β) denotes the following specialization xij −→ 1 if j < n, xin −→ t, ∀ i = 1, . . . , n− 1 of the polynomial Pn(xij ;β). In particular, Tn(k, k) = ( n−2 k−1 ) and Tn(k, 1) = T (n− 2, k − 1), where T (n, k) := 1 k + 1 ( n+ k k )( n k ) On Some Quadratic Algebras 103 is equal to the number of Schröder paths (i.e., consisting of steps U = (1, 1), D = (1,−1), H = (2, 0) and never going below the x-axis) from (0, 0) to (2n, 0), having k U ’s, see [131, A088617]. Moreover, Tn(n− 1, r) = Tab(n− 2, r − 1), where Tab(n, k) := k + 1 n+ 1 ( 2n− k n ) = F (2) n−k(k) is equal to the number of standard Young tableaux of the shape (n, n−k), see [131, A009766]. Recall that F (p) n (b) = 1+b n ( np+b n−1 ) stands for the generalized Fuss–Catalan number. (3) After the specialization xij −→ 1 the polynomial P (xij) is transformed to the polynomial Pn(β) := n−1∑ k=0 N(n, k)(1 + β)k, where N(n, k) := 1 n ( n k )( n k+1 ) , k = 0, . . . , n− 1, stand for the Narayana numbers. Furthermore, Pn(β) = n−1∑ d=0 sn(d)βd, where sn(d) = 1 n+ 1 ( 2n− d n )( n− 1 d ) is the number of ways to draw n − 1 − d diagonals in a convex (n + 2)-gon, such that no two diagonals intersect their interior. Therefore, the number of (nonzero) terms in the polynomial P (xij ;β) is equal to the n-th little Schröder number sn := n−1∑ d=0 sn(d), also known as the n-th super-Catalan number, see, e.g., [131, A001003]. (4) Upon the specialization x1j −→ t, 1 ≤ j ≤ n, and that xij −→ 1 if 2 ≤ i < j ≤ n, the polynomial P (xij ;β) is transformed to the polynomial Pn(β, t) = t n∑ k=1 (1 + β)n−k ∑ π tp(π), where the second summation runs over the set of Dick paths π of length 2n with exactly k picks (UD-steps), and p(π) denotes the number of valleys (DU-steps) that touch upon the line x = 0. (5) The polynomial P (xij ;β) is invariant under the action of anti-involution φ ◦ τ , see [72, Section 5.1.1] for definitions of φ and τ . (6) Follow [133, Exercise 6.C8(c)] consider the specialization xij −→ ti, 1 ≤ i < j ≤ n, and def ine Pn(t1, . . . , tn−1;β) = Pn(xij = ti;β). One can show, cf. [133], that Pn(t1, . . . , tn−1;β) = ∑ βn−kti1 · · · tik , where the sum runs over all pairs {(a1, . . . , ak), (i1, . . . , ik) ∈ Z≥1 × Z≥1} such that 1 ≤ a1 < a2 < · · · < ak, 1 ≤ i1 ≤ i2 · · · ≤ ik ≤ n and ij ≤ aj for all j. 104 A.N. Kirillov Now we are ready to state our main result about polynomials Pn(t1, . . . , tn;β). Let π := πn ∈ Sn be the permutation π = ( 1 2 3 . . . n 1 n n− 1 · · · 2 ) . Then Pn(t1, . . . , tn−1;β) = ( n−1∏ i=1 tn−ii ) G(β) π ( t−1 1 , . . . , t−1 n−1 ) = ∑ T wt(T ), (5.1) where G (β) w (x1, . . . , xn−1) denotes the β-Grothendieck polynomial corresponding to a permuta- tion w ∈ Sn, see [42] or Appendix A.1; summation in the right hand side of the second equality runs over the set of all dissections T of a convex (n + 2)-gon, and wt(T ) denotes weight of a dissection T , namely, wt(T ) = ∏ d∈T xdβ n−3−\|T \|, where the product runs over diagonals in T , xd = xij , if diagonal d connects vertices i and j, i < j, and \|T \| denotes the number of diagonals in dissection T . In particular, G(β) π (x1 = 1, . . . , xn−1 = 1) = n−1∑ k=0 N(n, k)(1 + β)k, where N(n, k) denotes the Narayana numbers, see item (3) of Proposition 5.4. More generally, write Pn(t, β) = ∑ k P (k) n (β)tk. Then G(β) π (x1 = t, xi = 1, ∀ i ≥ 2) = n−1∑ k=0 P (k) n−1 ( β−1 ) βktn−1−k. Comments 5.5. • Note that if β = 0, then one has G (β=0) w (x1, . . . , xn−1) = Sw(x1, . . . , xn−1), that is the β-Grothendieck polynomial at β = 0, is equal to the Schubert polynomial corresponding to the same permutation w. Therefore, if π = ( 1 2 3 . . . n 1 n n− 1 . . . 2 ) , then Sπ(x1 = 1, . . . , tn−1 = 1) = Cn−1, (5.2) where Cm denotes the m-th Catalan number. Using the formula (5.2) it is not difficult to check that the following formula for the principal specialization of the Schubert polynomial Sπ(Xn) is true Sπ ( 1, q, . . . , qn−1 ) = q( n−1 3 )Cn−1(q), (5.3) where Cm(q) denotes the Carlitz–Riordan q-analogue of the Catalan numbers, see, e.g., [134]. The formula (5.3) has been proved in [44] using the observation that π is a vex- illary permutation, see [92] for the a definition of the latter. A combinatorial/bijective proof of the formula (5.2) is due to A. Woo [142]. On Some Quadratic Algebras 105 • The Grothendieck polynomials, had been defined originally by A. Lascoux and M.-P. Schüt- zenberger, see, e.g., [86], correspond to the case β = −1. In this case Pn(−1) = 1 if n ≥ 0, and therefore the specialization G (−1) w (x1 = 1, . . . , xn−1 = 1) = 1 for all w ∈ Sn. • In Section 5.2.2, Theorems 5.28 and 5.29, we state a generalization of the second equality in the formula (5.1) to the case of Richardson’s permutations of the form 1k×w(n−k) 0 := π (n) k , and relate monomials which appear in a combinatorial formula51 for the corresponding β- Grothendieck polynomial, and/with the set of k-dissections and k-triangulations of a con- vex (n+k+1)-gon, and the Lagrange inversion formula, see Section 5.4.2 for more details. Clearly, the Richardson permutations π (0) k are special subset of permutations of the form 1k × wλ := w (λ) k , where wλ stands for the dominant permutation of shape λ. An analogue and extension of the first equality in the formula (5.1) for permutations of the form w (λ) 1 has been proved in [39, Theorem 5.4]. We state here a particular case of that result related with the Fuss– Catalan numbers obtained independently by the author of the present paper as a generalization of [133, Exercise 8C5(c)] and [142] to the case of Fuss–Catalan numbers. Namely, let λ = (λ1, . . . , λk = 1) be a Young diagram such that λi − λi+1 ≤ 1. Therefore, the boundary ∂(λ) of λ, that is the set of the last boxes in each row of λ, is a disjoint union of vertical intervals. To the last box of the lowermost interval we attach the generator x23. To the next box of that interval, if exists, we attach the generator u24 and so on, up to the top box of that interval is equipped with the generator, say x2,k1 . It is clear that k1 = λ′1 − λ′2 + 2. Now let us consider the next vertical interval. To the bottom box of that interval we attach the variable xk1,k1+1, to the next box we attach the variable xk1,k1+2 and so on. Let the top of that vertical interval is equipped with the generator xk1,k2 ; it is clear that k2 = λ′1 − λ′3 + 2. Applying this procedure successively step by step to each vertical interval, we attach the variable ub to each box b in the boundary of Young diagram λ. Finally we attach the monomial Mλ = x12 ∏ b∈∂(λ) xb. Theorem 5.6 ([39]). Let λ be a partition such that λi−λi+1 ≤ 1, ∀ i ≥ 1, and set N := λ′1 + 2. Let wλ ∈ SN be a unique dominant partition of shape λ, and Mλ ∈ ÂCYBN (β) be the monomial associated with the boundary ∂(λ) of partition λ. Then PMλ (xij = ti, ∀ i, j;β) = tλG (β) 1×wλ ( t−1 1 , . . . , t−1 N ) , where tλ := t λ′1 1 · · · t λ′N N . In other words, after the specialization xij −→ t−1 i , ∀ i, j, the spe- cialized reduced polynomial corresponding to the monomial Mλ is equal to t−λ multiplied by the β-Grothendieck polynomial associated with the permutation 1× wλ. Let us illustrate the above theorem by example. We take λ = 43221. In this caseN = 7 = 5+2 and w := wλ = [1, 6, 5, 4, 7, 3, 2]. The monomial corresponding to the boundary of λ is equal to x12x23x34x35x56x67 ∈ ÂCYB7. Since the both reduced and β-Grothendieck polynomials appearing in this example are huge, we display only its specialized values at xij = 1, ∀ i, j and ti = 1, ∀ i. We set also d := β − 1. It is not difficult to check that the reduced polynomial corresponding to monomial x12x23x34x35x56 after the specialization xij = 1, ∀ 1 ≤ i < j ≤ 5, and the identification xi,6 = x1,6, 1 ≤ i ≤ 5, is equal to (9, 20, 14, 3)βx16 + (9, 15, 6)βx 2 16 + (4, 4)βx 3 16 + x4 16. 51See [13, 44, 77] for example. 106 A.N. Kirillov Finally after multiplication of the above expression by x67, applying 3-term relations (b) in the algebra ÂCYB7 to the result obtained,and and taking the specialization xi,7 = 1, ∀ i, we will come to the following expression (9, 20, 14, 3)β(2 + β) + (9, 15, 6)β(3 + 2β)2 + (4, 4)β(4 + 3β) + (5 + 4β) = (66, 144, 108, 32, 3)β. One can check that the latter polynomial is equal to Gβ w(1). Corollary 5.7 (monomials and Fuss–Catalan numbers FC (p+1) n ). Let p, n, b be integers, consider diagram λ = (nb, (n − 1)p, (n − 2)p, . . . , 2p, 1p) and dominant permutation w ∈ S(n−1)p+b+2 of shape λ. Let us define monomial Mn,p,b = x12 n−2∏ j=0 ( p+2∏ a=3 xjp+2,jp+a ) b+2∏ a=3 x(n−1)p+2,(n−1)p+a. Then PMn,p,b (xij = 1, ∀ i, j)(β) = n∑ k=1 1 k ( n− 1 k − 1 )( pn− b k − 1 ) (β + 1)k−1. Moreover, PMn,p,b (xij = 1, ∀ i, j)(β = 0) = 1 np− b+ 1 ( n(p+ 1)− b n ) = 1 n ( n(p+ 1)− b n− 1 ) , where b := b− 1−(−1)b 2 . For b = 0 the right hand side of the above equality is equal to the Fuss–Narayana poly- nomial, see Theorem 5.46 and Proposition 5.47; a combinatorial interpretation of the value PMn,p,b (xij = 1, ∀ i, j)(β = 1) one can find in [110]. Note that reduced expressions for monomial Mn,p,b in the (noncommutative) algebra ÂCYBn(β) up to applying the commutativity rules (a), Definition 5.1, is unique. It seems an interesting problem to construct a natural bijection between the set of monomials in the (noncommutative) reduced expression associated with monomials Mn,p,0 and the set of (p+ 1)-gulations52 Finally we remark that there are certain connections of the β-Grothendieck polynomials corresponding to shifted dominance permutations (i.e., permutations of the form 1k×wλ) and some generating functions for the set of bounded by k plane partitions of shape λ, see, e.g., [44]. In the case of a staircase shape partition λ = (n − 1, . . . , 1) one can envision (cf. [128, 135]) a connection/bijection between the set of k-bounded plane partitions of that shape and k-dissections of a convex (n + k + 1)-gon. However in the case k ≥ 2 it is not clear does there exist a monomial M in the algebra ÂCYBn such that the value of the corresponding reduced polynomial at xij = 1, ∀ i, j is equal to the number of k-dissections (k ≥ 2) of a convex (n+ k + 1)-gon. Exercises 5.8. (1) (a) Let as before, π = ( 1 2 3 . . . n 1 n n− 1 . . . 2 ) . 52That is the set of dissections of a convex pk-gon by (maximal) collection of non-crossing diagonals such that the all regions obtained are a convex (p+ 2)-gons of a convex kp-gon. On Some Quadratic Algebras 107 Show that Sπ(x1 = q, xj = 1, ∀ j 6= i) = n−2∑ a=0 n− a− 1 n− 1 ( n+ a− 2 a ) qa. Note that the number n− k + 1 n+ 1 ( n+ k k ) is equal to the dimension of irreducible representation of the symmetric group Sn+k that corre- sponds to partition (n+ k, k). (b) Big Schröder numbers, paths and polynomials G (β) 1×w(n−1) 0 (x1 = q, xi = 1, ∀ i ≥ 2). Let n ≥ 1 and k ≥ 0 be integers, denote by Sk,n the number of big Schröder paths of type (k, n), that is lattice paths from the point (0, 0) and ending at point (2n + k, k), using only the steps U = (1, 1), H = (2, 0) and D = (1,−1) and never going below the line x = 0. The numbers S(n) := S0,n commonly known as big Schröder numbers, see, e.g., [131, A001003]. It is well- known that Sk,n = k + 1 n n∑ a=0 ( n a )( n+ k + a n− 1 ) . Show that G (β) 1×w(n−1) 0 (x1 = q, xi = 1, ∀ i ≥ 2) = n−2∑ k=0 Sk,n−2−k(β)qn−k−2, where Sk,n(β) is the generating functions of the big Schröder paths of type (k, n) according to the number of horizontal steps H. (c) Show that the polynomial G (β) 1×w(n−1−1) 0 (x1 = q, xi = 1, ∀ i ≥ 2) belongs to the ring N[q, β + 1]. For example, for n = 8 one has G (β) 1×w(7) 0 (x1 = q, xi = 1, ∀ i ≥ 2) = (0, 1, 15, 50, 50, 15, 1)β+1t 6 + (0, 2, 24, 60, 40, 6)β+1t 5 + (0, 3, 27, 45, 15)β+1t 4 + (0, 4, 24, 20)β+1t 3 + (0, 5, 15)β+1t 2 + 6(β + 1)t+ 1. Show that Sk,n(β) = k + 1 n n∑ a=0 ( n a )( n+ k + a n− 1 ) βn−a = k + 1 k + 1 + n ( 2n+ k n ) + · · ·+ ( n+ k n ) βn. (d) Write G (β) 1k×w(n−k) 0 (x1 = q, xi = 1, ∀ i ≥ 2) = Ak,n(β)qn−k−1 + · · ·+Bn,k(β). Show that Ak,n = (1 + β)kG (β) k,n−1(xi = 1, ∀ i ≥ 1), Bk,n = G (β) k−1,n−1(xi = 1, ∀ i ≥ 1). (2) Consider the commutative quotient ÃCYB ab n (α, β) of the algebra ÃCYBn(α, β), i.e., as- sume that the all generators {xij \| 1 ≤ i < j ≤ n are mutually commute. Denote by Pn(xij ;α, β) 108 A.N. Kirillov the image of polynomial the Pn(xij ;α, β) ∈ ÃCYBn(α, β) in the algebra ÃCYB ab n (α, β). Finally, define polynomials Pn(t, α, β) to be the specialization xij −→ 1 if j < n, xin −→ t if 1 ≤ i < n. Show that (a) Polynomial Pn(t, α, β) does not depend on on order in which relations (a) and (b), see Definition 5.1, have been applied. (b) Pn(1, α = 1, β = 0) = ∑ k≥0 (2n− 2k)! k!(n+ 1− k)!(n− 2k)! , see [131, A052709(n)] for combinatorial interpretations of these numbers. For example, P7(t, α, β) = t7 + 6(1 + β)t6 + [(0, 5, 15)β+1 + 6α]t5 + [(0, 4, 24, 20)β+1 + α(5, 29)β+1]t4 + [(0, 3, 27, 45, 15)β+1 + α(4, 45, 55)β+1 + 14α2]t3 + [(0, 2, 24, 60, 40, 6)β+1 + α(3, 48, 115, 50)β+1 + α2(21, 49)β+1]t2 + [(0, 1, 15, 50, 50, 15, 1)β+1 + α(2, 38, 130, 110, 20)β+1 + α2(21, 91, 56)β+1 + 14α3]t+ α(1, 15, 50, 50, 15, 1)β+1 + α2(14, 70, 70, 14)β+1 + α3(21, 21)β+1. (c) Show that in fact Pn(1, α, 0) = ∑ k≥0 1 n+ 1 ( 2n− 2k n )( n+ 1 k ) αk = ∑ k≥0 Tn+2(n− k, k + 1) 2n− 1− 2k αk, see Proposition 5.4(2), for definition of numbers Tn(k, r). As for a combinatorial interpretation of the polynomials Pn(1, α, 0), see [131, A117434, A085880]. (3) Consider polynomials Pn(t, β) as it has been defined in Proposition 5.4(2). Show that Pn(t, β) = Pn(t, α = 0, β) = tn + n−1∑ r=1 tr ( n−1−r∑ k=0 r n ( n k + r )( n− r − 1 k ) (1 + β)n−r−k ) , cf., e.g., [131, A033877]. A few comments in order. Several combinatorial interpretations of the integer numbers Un(r, k) := r n+ 1 ( n+ 1 k + r )( n− r k ) are well-known. For example, if r = 1, the numbers Un(1, k) = 1 n ( n k+1 )( n k ) are equal to the Narayana numbers, see, e.g., [131, A001263]; if r = 2, the number Un(2, k) counts the number of Dyck (n+ 1)-paths whose last descent has length 2 and which contain n− k peaks, see [131, A108838] for details. Finally, it’s easily seen, that Pn(1, β) = A127529(n), and Pn(t, 1) = A033184(n), see [131]. (4) Show that Pn(t, α, β) ∈ N[t, α][β + 1], that is the polynomial Pn(t, α, β) is a polynomial of β+ 1 with coefficients from the ring N[t, α]. On Some Quadratic Algebras 109 Show that Pn(0, 1, β) ∈ N[β + 2]. For example, P7(0, 1, β) = (0, 3, 8, 14, 10, 1)β+2, P8(0, 1, β) = (1, 3, 11, 25, 35, 15, 1)β+2. Show that [131] Pn(1, 1, 0) = A052709(n+ 1), that is the number of underdiagonal lattice paths from (0, 0) to (n−1, n−1) and such that each step is either (1, 0), (0, 1), or (2, 1). For example, P7(1, 1, 0) = 1697 = A052709(8). Cf. with the next exercise. Show that [131] Pn(0, 1, 0) = A052705(n), namely, the number of underdiagonal paths from (0,0) to the line x = n − 2, using only steps (1, 0), (0, 1) and NE = (2, 1). For example, P7(0, 1, 0) = 36 + 106 + 120 + 64 + 15 + 1 = 342 = A052705(7). Show that [131] ∂ ∂a Pn(a, b = 1,β = 0,α = 1,y = z = 1) = A005775, that is the number number of paths in the half-plane x ≥ 0 from (0, 0) to (n−1, 2) or (n−1,−3), and consisting of steps U = (1, 1), D = (1,−1) and H = (1, 0) that contain at least one UUU but avoid UUU ′s starting above level 0. 5.1.1 Multiparameter deformation of Catalan, Narayana and Schröder numbers Let b = (β1, . . . , βn−1) be a set of mutually commuting parameters. We define a multiparameter analogue of the associative quasi-classical Yang–Baxter algebra M̂ACYBn(b) as follows. Definition 5.9 (cf. Definition 2.20). The multiparameter associative quasi- classical Yang– Baxter algebra of weight b, denoted by M̂ACYBn(b), is an associative algebra, over the ring of polynomials Z[β1, . . . , βn−1], generated by the set of elements {xij , 1 ≤ i < j ≤ n}, subject to the set of relations (a) xijxkl = xklxij if {i, j} ∩ {k, l} = ∅, (b) xijxjk = xikxij + xjkxik + βixik if 1 ≤ 1 < i < j ≤ n. Consider the “Coxeter element” wn ∈ M̂ACYBn(b) which is equal to the ordered product of “simple generators”: wn := n−1∏ a=1 xa,a+1. Now we can use the same method as in [133, Exercise 8.C5(c)], see Section 5.1, to define the reduced form of the Coxeter element wn. Namely, let us bring the element wn to the reduced form in the algebra M̂ACYBn(b), that is, let us consecutively apply the defining relations (a) and (b) to the element wn in any order until unable to do so. Denote the resulting (noncommutative) polynomial by P (xij ; b). In principal, the polynomial itself can depend on the order in which the relations (a) and (b) are applied. 110 A.N. Kirillov Proposition 5.10 (cf. [133, Exercise 8.C5(c)], [99, 100]). The specialized polynomial P (xij = 1, ∀ i, j, b) does not depend on the order in which relations (a) and (b) have been applied. To state our main result of this subsection, let us define polynomials Q(β1, . . . , βn−1) := P (xij = 1, ∀ i, j; β1 − 1, β2 − 1, . . . , βn−1 − 1). Example 5.11. Q(β1, β2) = 1 + 2β1 + β2 + β2 1 , Q(β1, β2, β3) = 1 + 3β1 + 2β2 + β3 + 3β2 1 + β1β2 + β1β3 + β2 2 + β3 1 , Q(β1, β2, β3, β4) = 1 + 4β1 + 3β2 + 2β3 + β4 + β1(6β1 + 3β2 + 3β3 + 2β4) + β2(3β2 + β3 + β4) + β2 3 + β2 1(4β1 + β2 + β3 + β4) + β1(β2 2 + β2 3) + β3 2 + β4 1 . Theorem 5.12. Polynomial Q(β1, . . . , βn−1) has non-negative integer coefficients. It follows from [133] and Proposition 5.4, that Q(β1, . . . , βn−1) ∣∣ β1=1,...,βn−1=1 = Catn. Polynomials Q(β1, . . . , βn−1) and Q(β1 +1, . . . , βn−1 +1) can be considered as a multiparameter deformation of the Catalan and (small) Schröder numbers correspondingly, and the homogeneous degree k part of Q(β1, . . . , βn−1) as a multiparameter analogue of Narayana numbers. 5.2 Grothendieck and q-Schröder polynomials 5.2.1 Schröder paths and polynomials Definition 5.13. A Schröder path of the length n is an over diagonal path from (0, 0) to (n, n) with steps (1, 0), (0, 1) and steps D = (1, 1) without steps of type D on the diagonal x = y. If p is a Schröder path, we denote by d(p) the number of the diagonal steps resting on the path p, and by a(p) the number of unit squares located between the path p and the diagonal x = y. For each (unit) diagonal step D of a path p we denote by i(D) the x-coordinate of the column which contains the diagonal step D. Finally, define the index i(p) of a path p as the some of the numbers i(D) for all diagonal steps of the path p. Definition 5.14. Define q-Schröder polynomial Sn(q;β) as follows Sn(q;β) = ∑ p qa(p)+i(p)βd(p), (5.4) where the sum runs over the set of all Schröder paths of length n. Example 5.15. S1(q;β) = 1, S2(q;β) = 1 + q + βq, S3(q;β) = 1 + 2q + q2 + q3 + β ( q + 2q2 + 2q3 ) + β2q3, S4(q;β) = 1 + 3q + 3q2 + 3q3 + 2q4 + q5 + q6 + β ( q + 3q2 + 5q3 + 6q4 + 3q5 + 3q6 ) + β2 ( q3 + 2q4 + 3q5 + 3q6 ) + β3q6. On Some Quadratic Algebras 111 Comments 5.16. The q-Schröder polynomials defined by the formula (5.4) are different from the q-analogue of Schröder polynomials which has been considered in [19]. It seems that there are no simple connections between the both. Proposition 5.17 (recurrence relations for q-Schröder polynomials). The q-Schröder polyno- mials satisfy the following relations Sn+1(q;β) = ( 1 + qn + βqn ) Sn(q;β) + k=n−1∑ k=1 ( qk + βqn−k ) Sk(q; q n−kβ)Sn−k(q;β), and the initial condition S1(q;β) = 1. Note that Pn(β) = Sn(1;β) and in particular, the polynomials Pn(β) satisfy the following recurrence relations Pn+1(β) = (2 + β) Pn(β) + (1 + β) n−1∑ k=1 Pk(β) Pn−k(β). (5.5) Theorem 5.18 (evaluation of the Schröder–Hankel determinant). Consider permutation π (n) k = ( 1 2 . . . k k + 1 k + 2 . . . n 1 2 . . . k n n− 1 . . . k + 1 ) . Let as before Pn(β) = n−1∑ j=0 N(n, j)(1 + β)j , n ≥ 1, (5.6) be Schröder polynomials. Then (1 + β)( k 2)G (β) π (n) k (x1 = 1, . . . , xn−k = 1) = Det \|Pn+k−i−j(β)\|1≤i,j≤k. Proof is based on an observation that the permutation π (n) k is a vexillary one and the recur- rence relations (5.5). Comments 5.19. (1) In the case β = 0, i.e., in the case of Schubert polynomials, Theorem 5.18 has been proved in [44]. (2) In the cases when β = 1 and 0 ≤ n − k ≤ 2, the value of the determinant in the r.h.s. of (5.6) is known53. One can check that in the all cases mentioned above, the formula (5.6) gives the same results. (3) Grothendieck and Narayana polynomials. It follows from the expression (5.5) for the Narayana–Schröder polynomials that Pn(β − 1) = Nn(β), where Nn(β) := n−1∑ j=0 1 n ( n j )( n j + 1 ) βj , denotes the n-th Narayana polynomial. Therefore, Pn(β−1) = Nn(β) is a symmetric polynomial in β with non-negative integer coefficients. Moreover, the value of the polynomial Pn(β − 1) at β = 1 is equal to the n-th Catalan number Cn := 1 n+1 ( 2n n ) . 53See, e.g., [19], or M. Ichikawa talk “Hankel determinants of Catalan, Motzkin and Schröder numbers and its q-analogue”, http://www.uec.tottori-u.ac.jp/~mi/talks/kyoto07.pdf. http://www.uec.tottori-u.ac.jp/~mi/talks/kyoto07.pdf 112 A.N. Kirillov It is well-known, see, e.g., [136], that the Narayana polynomial Nn(β) is equal to the gene- rating function of the statistics π(p) = (number of peaks of a Dick path p)− 1 on the set Dickn of Dick paths of the length 2n Nn(β) = ∑ p βπ(p). Moreover, using the Lindström–Gessel–Viennot lemma54, one can see that DET \|Nn+k−i−j(β)\|1≤i,j≤k = β(k2) ∑ (p1,...,pk) βπ(p1)+···+π(pk), (5.7) where the sum runs over k-tuple of non-crossing Dick paths (p1, . . . , pk) such that the path pi starts from the point (i− 1, 0) and has length 2(n− i+ 1), i = 1, . . . , k. We denote the sum in the r.h.s. of (5.7) by N (k) n (β). Note that N (k) k−1(β) = 1 for all k ≥ 2. Thus, N (k) n (β) is a symmetric polynomial in β with non-negative integer coefficients, and N(k) n (β = 1) = C(k) n = ∏ 1≤i≤j≤n−k 2k + i+ j i+ j = ∏ 2a≤n−k−1 ( 2n−2a 2k )( 2k+2a+1 2k ) . As a corollary we obtain the following statement Proposition 5.20. Let n ≥ k, then G (β−1) π (n) k (x1 = 1, . . . , xn = 1) = N(k) n (β). Summarizing, the specialization G (β−1) π (n) k (x1 = 1, . . . , xn = 1) is a symmetric polynomial in β with non-negative integer coefficients, and coincides with the generating function of the statistics k∑ i=1 π(pi) on the set k-Dickn of k-tuple of non-crossing Dick paths (p1, . . . , pk). Example 5.21. Take n = 5, k = 1. Then π (5) 1 = (15432) and one has G (β) π (5) 1 ( 1, q, q2, q3 ) = q4(1, 3, 3, 3, 2, 1, 1) + q5(1, 3, 5, 6, 3, 3)β + q7(1, 2, 3, 3)β2 + q10β3. It is easy to compute the Carlitz–Riordan q-analogue of the Catalan number C5, namely, C5(q) = (1, 3, 3, 3, 2, 1, 1). Remark 5.22. The value Nn(4) of the Narayana polynomial at β = 4 has the following com- binatorial interpretation: Nn(4) is equal to the number of different lattice paths from the point (0, 0) to that (n, 0) using steps from the set Σ = {(k, k) or (k,−k), k ∈ Z>0}, that never go below the x-axis, see [131, A059231]. Exercises 5.23. (a) Show that γk,n := C (k+1) n C (k) n = (2n− 2k)!(2k + 1)! (n− k)!(n+ k + 1)! . 54See, e.g., https://en.wikipedia.org/wiki/Lindstrom-Gessel-Viennot_lemma. https://en.wikipedia.org/wiki/Lindstrom-Gessel-Viennot_lemma On Some Quadratic Algebras 113 (b) Show that γk,n ≤ 1 if k ≤ n ≤ 3k + 1, and γk,n ≥ 2n−3k−1 if n > 3k + 1. (4) Polynomials Fw(β), Hw(β), Hw(q, t;β) and Rw(q;β). Let w ∈ Sn be a permutation, G (β) w (Xn) and G (β) w (Xn, Yn) be the corresponding β-Grothendieck and double β-Grothendieck polynomials. We denote by G (β) w (1) and by G (β) w (1; 1) the specializations Xn := (x1 = 1, . . ., xn = 1), Yn := (y1 = 1, . . . , yn = 1) of the β-Grothendieck polynomials introduced above. Theorem 5.24. Let w ∈ Sn be a permutation. Then (i) The polynomials Fw(β) := G (β−1) w (1) and Hw(β) := G (β−1) w (1; 1) have both non-negative integer coefficients. (ii) One has Hw(β) = (1 + β)`(w)Fw ( β2 ) . (iii) Let w ∈ Sn be a permutation, define polynomials Hw(q, t;β) := G(β) w (x1 = q, x2 = q, . . . , xn = q, y1 = t, y2 = t, . . . , yn = t) to be the specialization {xi = q, yi = t, ∀ i} of the double β-Grothendieck polynomial G (β) w (Xn, Yn). Then Hw(q, t;β) = (q + t+ βqt)`(w)Fw((1 + βq)(1 + βt)). In particular, Hw(1, 1;β) = (2 + β)`(w)Fw((1 + β)2). (iv) Let w ∈ Sn be a permutation, define polynomial Rw(q;β) := G(β−1) w (x1 = q, x2 = 1, x3 = 1, . . .) to be the specialization {x1 = q, xi = 1, ∀ i ≥ 2}, of the (β − 1)-Grothendieck polynomial G (β−1) w (Xn). Then Rw(q;β) = qw(1)−1Rw(q;β), where Rw(q;β) is a polynomial in q and β with non-negative integer coefficients, and Rw(0;β = 0) = 1. (v) Consider permutation w (1) n := [1, n, n − 1, n − 2, . . . , 3, 2] ∈ Sn. Then H w (1) n (1, 1; 1) = 3(n−1 2 )Nn(4). In particular, if w (k) n = (1, 2, . . . , k, n, n− 1, . . . , k + 1) ∈ Sn, then S (β−1) w (k) n (1; 1) = (1 + β)( n−k 2 )S (β−1) w (k) n ( β2 ) . See Remark 5.22 for a combinatorial interpretation of the number Nn(4). Example 5.25. Consider permutation v = [2, 3, 5, 6, 8, 9, 1, 4, 7] ∈ S9 of the length 12, and set x := (1 + βq)(1 + βt). One can check that Hv(q, t;β) = x12(1 + 2x) ( 1 + 6x+ 19x2 + 24x3 + 13x4 ) , and Fv(β) = (1 + 2β)(1 + 6β + 19β2 + 24β3 + 13β4). Note that Fv(β = 1) = 27 × 7, and 7 = AMS(3), 26 = CSTCTPP(3), cf. Conjecture 5.52, Section 5.2.4. 114 A.N. Kirillov Remark 5.26. One can show, cf. [92, p. 89], that if w ∈ Sn, then Rw(1, β) = Rw−1(1, β). However, the equality Rw(q, β) = Rw−1(q, β) can be violated, and it seems that in general, there are no simple connections between polynomials Rw(q, β) and Rw−1(q, β), if so. From this point we shell use the notation (a0, a1, . . . , ar)β := r∑ j=0 ajβ j , etc. Example 5.27. Let us take w = [1, 3, 4, 6, 7, 9, 10, 2, 5, 8]. Then Rw(q, β) = (1, 6, 21, 36, 51, 48, 26)β + qβ(6, 36, 126, 216, 306, 288, 156)β + q2β3(20, 125, 242, 403, 460, 289)β + q3β5(6, 46, 114, 204, 170)β. Moreover, Rw(q, 1) = (189, 1134, 1539, 540)q. On the other hand, w−1 = [1, 8, 2, 3, 9, 4, 5, 10, 6, 7], and Rw−1(q, β) = (1, 6, 21, 36, 51, 48, 26)β + qβ(1, 6, 31, 56, 96, 110, 78)β + q2β(1, 6, 27, 58, 92, 122, 120, 78)β + q3β(1, 6, 24, 58, 92, 126, 132, 102, 26)β + q4β(1, 6, 22, 57, 92, 127, 134, 105, 44)β + q5β(1, 6, 21, 56, 91, 126, 133, 104, 50)β + q6β(1, 6, 21, 56, 91, 126, 133, 104, 50)β. Moreover, Rw−1(q, 1) = (189, 378, 504, 567, 588, 588, 588)q. Notice that w = 1× u, where u = [2, 3, 5, 6, 8, 9, 1, 4, 7]. One can show that Ru(q, β) = (1, 6, 11, 16, 11)β + qβ2(10, 20, 35, 34)β + q2β4(5, 14, 26)β. On the other hand, u−1 = [7, 1, 2, 8, 3, 4, 9, 5, 6] and Ru−1(1, β) = (1, 6, 21, 36, 51, 48, 26)β = Ru(1, β). Recall that by our definition (a0, a1, . . . , ar)β := r∑ j=0 ajβ j . 5.2.2 Grothendieck polynomials and k-dissections Let k ∈ N and n ≥ k − 1, be a integer, define a k-dissection of a convex (n + k + 1)-gon to be a collection E of diagonals in (n + k + 1)-gon not containing (k + 1)-subset of pairwise crossing diagonals and such that at least 2(k − 1) diagonals are coming from each vertex of the (n+k+ 1)-gon in question. One can show that the number of diagonals in any k-dissection E of a convex (n+k+ 1)-gon contains at least (n+k+ 1)(k−1) and at most n(2k−1)−1 diagonals. We define the index of a k-dissection E to be i(E) = n(2k − 1)− 1−#\|E\|. Denote by T (k) n (β) = ∑ E βi(E) the generating function for the number of k-dissections with a fixed index, where the above sum runs over the set of all k-dissections of a convex (n+ k + 1)-gon. Theorem 5.28. G (β) π (n) k (x1 = 1, . . . , xn = 1) = T (k) n (β). On Some Quadratic Algebras 115 Mopre generally, let n ≥ k > 0 be integers, consider a convex (n + k + 1)-gon Pn+k+1 and a vertex v0 ∈ Pn+k+1. Let us label clockwise the vertices of Pn+k+1 by the numbers 1, 2, . . . , n+k+1 starting from the vertex v0. Let Dis(Pn+k+1) denotes the set of all k-dissections of the (n+ k+ 1)-gon Pn+k+1. We denote by D0 := Dis0(Pn+k+1) the “minimal” k-dissection of the (n+ k + 1)-gon Pn+k+1 in question consisting of the set of diagonals connecting vertices va and va+r, where 2 ≤ r ≤ k, 1 ≤ a ≤ n + k + 1, and for any positive integer a we denote by a a unique integer such that 1 ≤ a ≤ n + k + 1 and a ≡ a(mod (n + k + 1)). For example, if k = 1, then Dis0(Pn+2) = ∅; if k = 3 and n = 4, in other words, P8 is a octagon, the minimal 3-dissection consists of 16 diagonals connecting vertices with the following labels 1→ 3→ 5→ 7→ 9 = 1, 2→ 4→ 6→ 8→ 10 = 2, 1→ 4→ 7→ 10 = 2→ 5→ 8→ 11 = 3→ 6→ 9 = 1. Now let D ∈ Dis(Pn+k+1) be a dissection. Consider a diagonal dij ∈ (D\D0), i < j which connects vertex vi with that vj . We attach variable xi to the diagonal dij in question and consider the following expression TPn+k+1 (Xn+k+1) = ∑ D∈Dis(Pn+k+1) β#\|D\D0\| ∑ dij∈(D\D0) i<j ∏ xi. Theorem 5.29. One has TPn+n+1(Xn+k+1) = βk(n−k) n∏ a=1 xmin(n−a+1,n−k) a Gβ−1 wnk ( x−1 1 , . . . , x−1 n ) . Exercises 5.30. It is not difficult to check that Gβ 15432(X5) = β3x3 1x 3 2x 2 3x4 + β2(x3 1x 3 2x3 + 2x3 1x 3 2x3x4 + 3x3 1x 2 2x 2 3x4 + 3x2 1x 3 2x 2 3x4) + β(x3 1x 3 2x3 + x3 1x 3 2x4 + 2x3 1x 2 2x3 + 2x2 1x 3 2x 2 3 + 3x3 1x 2 2x3x4 + 3x3 1x2x 2 3x4 + 3x2 1x 3 2x3x4 + 3x2 1x 2 2x 2 3x4 + 3x1x 3 2x 2 3x4) + x3 1x 2 2x3 + x3 1x 2 2x4 + x3 1x2x 2 3 + x3 1x2x3x4 + x3 1x 2 3x4 + x2 1x 3 2x3 + x2 1x 3 2x4 + x2 1x 2 2x 2 3 + x2 1x 2 2x3x4 + x2 1x2x 2 3x4 + x1x 3 2x 2 3 + x1x 3 2x3x4 + x1x 2 2x 2 3x4 + x3 2x 2 3x4. Describe bijection between dissections of hexagon P6 (the case k = 1, n = 4) and the above listed monomials involved in the β-Grothendieck polynomial Gβ 15432(x1, x2, x3, x4). A k-dissection of a convex (n+k+1)-gon with the maximal number of diagonals (which is equal to n(2k−1)−1) is called k-triangulation. It is well-known that the number of k-triangulations of a convex (n+k+1)-gon is equal to the Catalan–Hankel number C (k) n−1. Explicit bijection between the set of k-triangulations of a convex (n + k + 1)-gon and the set of k-tuple of non-crossing Dick paths (γ1, . . . , γk) such that the Dick path γi connects points (i− 1, 0) and (2n− i− 1, 0), has been constructed in [128, 135]. 5.2.3 Grothendieck polynomials and q-Schröder polynomials Let π (n) k = 1k × w(n−k) 0 ∈ Sn be the vexillary permutation as before, see Theorem 5.18. Recall that π (n) k = ( 1 2 . . . k k + 1 k + 2 . . . n 1 2 . . . k n n− 1 . . . k + 1 ) . 116 A.N. Kirillov (A) Principal specialization of the Schubert polynomial S π (n) k . Note that π (n) k is a vexillary permutation of the staircase shape λ = (n−k−1, . . . , 2, 1) and has the staircase flag φ = (k + 1, k + 2, . . . , n − 1). It is known, see, e.g., [92, 139], that for a vexillary permutation w ∈ Sn of the shape λ and flag φ = (φ1, . . . , φr), r = `(λ), the corresponding Schubert polynomial Sw(Xn) is equal to the multi-Schur polynomial sλ(Xφ), where Xφ denotes the flagged set of variables, namely, Xφ = (Xφ1 , . . . , Xφr) and Xm = (x1, . . . , xm). Therefore we can write the following determinantal formula for the principal specialization of the Schubert polynomial corresponding to the vexillary permutation π (n) k S π (n) k ( 1, q, q2, . . . ) = DET ([ n− i+ j − 1 k + i− 1 ] q ) 1≤i,j≤n−k , where [ n k ] q denotes the q-binomial coefficient. Let us observe that the Carlitz–Riordan q-analogue Cn(q) of the Catalan number Cn is equal to the value of the q-Schröder polynomial at β = 0, namely, Cn(q) = Sn(q, 0). Lemma 5.31. Let k, n be integers and n > k, then (1) DET ([ n− i+ j − 1 k + i− 1 ] q ) 1≤i,j≤n−k = q( n−k 3 )C(k) n (q), (2) DET ( Cn+k−i−j(q) ) 1≤i,j≤k = qk(k−1)(6n−2k−5)/6 C(k) n (q). (B) Principal specialization of the Grothendieck polynomial G (β) π (n) k . Theorem 5.32. q( n−k+1 3 )−(k−1)(n−k2 ) DET ∣∣Sn+k−i−j ( q; qi−1β )∣∣ 1≤i,j≤k = qk(k−1)(4k+1)/6 k−1∏ a=1 ( 1 + qa−1β ) G π (n) k ( 1, q, q2, . . . ) . Corollary 5.33. (1) If k = n− 1, then DET \|S2n−1−i−j ( q; qi−1β ) \|1≤i,j≤n−1 = q(n−1)(n−2)(4n−3)/6 n−2∏ a=1 ( 1 + qa−1β )n−a−1 , (2) If k = n− 2, then qn−2 DET ∣∣S2n−2−i−j ( q; qi−1β )∣∣ 1≤i,j≤n−2 = q(n−2)(n−3)(4n−7)/6 n−3∏ a=1 ( 1 + qa−1β )n−a−2 { (1 + β)n−1 − 1 β } . Generalization. Let n = (n1, . . . , np) ∈ Np be a composition of n so that n = n1 + · · ·+np. We set n(j) = n1 + · · ·+ nj , j = 1, . . . , p, n(0) = 0. Now consider the permutation w(n) = w (n1) 0 × w(n2) 0 × · · · × w(np) 0 ∈ Sn, where w (m) 0 ∈ Sm denotes the longest permutation in the symmetric group Sm. In other words, w(n) = ( 1 2 . . . n1 n(2) . . . n1 + 1 . . . n(p−1) . . . n n1 n1 − 1 . . . 1 n1 + 1 . . . n(2) . . . n . . . n(p−1)+1 ) . On Some Quadratic Algebras 117 For the permutation w(n) defined above, one has the following factorization formula for the Grothendieck polynomial corresponding to w(n) [92] G (β) w(n) = G (β) w (n1) 0 ×G (β) 1n1×w(n2) 0 ×G (β) 1n1+n2×w(n3) 0 × · · · ×G (β) 1n1+...np−1×w(np) 0 . In particular, if w(n) = w (n1) 0 × w(n2) 0 × · · · × w(np) 0 ∈ Sn, (5.8) then the principal specialization G (β) w(n) of the Grothendieck polynomial corresponding to the permutation w, is the product of q-Schröder–Hankel polynomials. Finally, we observe that from discussions in Section 5.2.1(3), Grothendieck and Narayana polynomials, one can deduce that G (β−1) w(n) (x1 = 1, . . . , xn = 1) = p−1∏ j=1 N (n(j)) n(j+1)(β). In particular, the polynomial G (β−1) w(n) (x1, . . . , xn) is a symmetric polynomial in β with non- negative integer coefficients. Example 5.34. (1) Let us take (non vexillary) permutation w = 2143 = s1s3. One can check that G(β) w (1, 1, 1, 1) = 3 + 3β + β2 = 1 + (β + 1) + (β + 1)2, and N4(β) = (1, 6, 6, 1), N3(β) = (1, 3, 1), N2(β) = (1, 1). It is easy to see that βG(β) w (1, 1, 1, 1) = DET ∣∣∣∣N4(β) N3(β) N3(β) N2(β) ∣∣∣∣ . On the other hand, DET ∣∣∣∣P4(β) P3(β) P3(β) P2(β) ∣∣∣∣ = (3, 6, 4, 1) = ( 3 + 3β + β2 ) (1 + β). It is more involved to check that q5(1 + β) G(β) w ( 1, q, q2, q3 ) = DET ∣∣∣∣ S4(q;β) S3(q;β) S3(q; qβ) S2(q; qβ) ∣∣∣∣ . (2) Let us illustrate Theorem 5.32 by a few examples. For the sake of simplicity, we consider the case β = 0, i.e., the case of Schubert polynomials. In this case Pn(q;β = 0) = Cn(q) is equal to the Carlitz–Riordan q-analogue of Catalan numbers. We are reminded that the q-Catalan– Hankel polynomials are defined as follows C(k) n (q) = qk(1−k)(4k−1)/6 DET \|Cn+k−i−j(q)\|1≤i,j≤n. In the case β = 0 the Theorem 5.32 states that if n = (n1, . . . , np) ∈ Np and the permutation w(n) ∈ Sn is defined by the use of (5.7), then Sw(n) ( 1, q, q2, . . . ) = q ∑ (ni3 )C (n1) n1+n2 (q)× C(n1+n2) n1+n2+n3 (q)× C(n−np) n (q). Now let us consider a few examples for n = 6. 118 A.N. Kirillov • n = (1, 5) =⇒ Sw(n)(1, q, . . .) = q10C (1) 6 (q) = C5(q). • n = (2, 4) =⇒ Sw(n)(1, q, . . .) = q4C (2) 6 (q) = DET ∣∣∣∣C6(q) C5(q) C5(q) C4(q) ∣∣∣∣. Note that Sw(2,4)(1, q, . . .) = Sw(1,1,4)(1, q, . . .). • n = (2, 2, 2) =⇒ Sw(n)(1, q, . . .) = C (2) 4 (q)C (4) 6 (q). • n = (1, 1, 4) =⇒ Sw(n)(1, q, . . .) = q4C (1) 2 (q)C (2) 4 (q) = q4C (2) 4 (q), the last equality follows from that C (k) k+1(q) = 1 for all k ≥ 1. • n = (1, 2, 3) =⇒ Sw(n)(1, q, . . .) = qC (1) 3 (q)C (3) 6 (q). • n = (3, 2, 1) =⇒ Sw(n)(1, q, . . .) = qC (3) 5 (q)C (5) 6 (q) = qC (3) 5 (q) = q(1, 1, 1, 1). Note that C (k) k+2(q) = [ k+1 1 ] q . Exercises 5.35. Let 1 ≤ k ≤ m ≤ n be integers, n ≥ 2k + 1. Consider permutation w = ( 1 2 . . . k k + 1 . . . n m m− 1 . . . m− k + 1 n . . . . . . 1 ) ∈ Sn. Show that Sw(1, q, . . .) = qn(D(w))C (m) n−m+k(q), where for any permutation w, n(D(w)) = ∑( di(w) 2 ) and di(w) denotes the number of boxes in the i-th column of the (Rothe) diagram D(w) of the permutation w, see [92, p. 8]. (C) A determinantal formula for the Grothendieck polynomials G (β) π (n) k . Define polynomials Φ(m) n (Xn) = n∑ a=m ea(Xn)βa−m, Ai,j(Xn+k−1) = 1 (i− j)! ( ∂ ∂β )j−1 Φ (n+1−i) k+n−i (Xk+n−i) if 1 ≤ i ≤ j ≤ n, and Ai,j(Xk+n−1) = i−j−1∑ a=0 en−i−a(Xn+k−i) ( i− j − 1 a ) if 1 ≤ j < i ≤ n. Theorem 5.36. DET \|Ai,j \|1≤i,j≤n = G (β) π (k) k+n (Xk+n−1). Comments 5.37. (a) One can compute the Grothendieck polynomials for yet another interesting family of permutations. namely, grassmannian permutations σ (n) k = ( 1 2 . . . k − 1 k k + 1 k + 2 . . . n+ k 1 2 . . . k − 1 n+ k k k + 1 . . . n+ k − 1 ) = sksk+1 · · · sn+k−1 ∈ Sn+k. On Some Quadratic Algebras 119 Then G (β) σk(n)(x1, . . . , xn+k) = k−1∑ j=0 s(n,1j)(Xk)β j , where s(n,1j)(Xk) denotes the Schur polynomial corresponding to the hook shape partition (n, 1j) and the set of variables Xk := (x1, . . . , xk). In particular, G (β) σk(n)(xj = 1, ∀ j) = ( n+ k − 1 k )k−1∑ j=0 k n+ j ( k − 1 j ) βj  = k−1∑ j=0 ( n+ j − 1 j ) (1 + β)j . (b) Grothendieck polynomials for grassmannian permutations. In the case of a grassmannian permutation w := σλ ∈ S∞ of the shape λ = (λ1 ≥ λ2 ≥ · · · ≥ λn) where n is a unique descent of w, one can prove the following formulas for the β-Grothendieck polynomial G(β) σλ (Xn) = DET ∣∣xλj+n−ji (1 + βxi) j−1 ∣∣ 1≤i,j≤n∏ 1≤i<j≤n (xi − xj) , (5.9) DET ∣∣h(β) λj+i,j (X[i,n]) ∣∣ 1≤i,j≤n = DET ∣∣h(β) λj+i,j (Xn) ∣∣ 1≤i,j≤n, where X[i,n] = (xi, xi+1, . . . , xn), and for any set of variables X h (β) n,k(X) = k−1∑ a=0 ( k − 1 a ) hn−k+a(X)βa, and hk(X) denotes the complete symmetric polynomial of degree k in the variables from the set X. A proof is a straightforward adaptation of the proof of special case β = 0 (the case of Schur polynomials) given by I. Macdonald [92, Section 2, equation (2.10) and Section 4, equation (4.9)]. Indeed, consider β-divided difference operators π (β) j , j = 1, . . . , n − 1, and π (β) w , w ∈ Sn, introduced in [42]. For example, π (β) j (f) = 1 xj − xj+1 ( (1 + βxj+1)f(Xn)− (1 + βxj)f(sj(Xn) ) . Now let w0 := w (n) 0 be the longest element in the symmetric group Sn. The same proves of the Statements 2.10, 2.16 from [92] show that π(β) w0 = a−1 δ w0 ∑ σ∈Sn (−1)`(σ) n−1∏ j=1 (1 + βxj) n−jσ  , where aδ = ∏ 1≤i<j≤n (xi − xj). On the other hand, the same arguments as in the proof of Statement 4.8 from [92] show that G(β) σλ (Xn) = π (β) w(0) ( xλ+δn ) . Application of the formula for operator π (β) w (0) n displayed above to the monomial xλ+δn finishes the proof of the first equality in (5.8). The statement that the right hand side of the equality (5.9) coincides with determinants displayed in the identity (5.9) can be checked by means of simple transformations. 120 A.N. Kirillov Problems 5.38. (1) Give a bijective prove of Theorem 5.28, i.e., construct a bijection between • the set of k-tuple of mutually non-crossing Schröder paths (p1, . . . , pk) of lengths (n, n− 1, . . . , n− k + 1) correspondingly, and • the set of pairs (m, T ), where T is a k-dissection of a convex (n+ k+ 1)-gon, and m is a upper triangle (0, 1)-matrix of size (k − 1) × (k − 1), which is compatible with natural statistics on the both sets. (2) Let w ∈ Sn be a permutation, and CS(w) be the set of compatible sequences corresponding to w, see, e.g., [13]. Define statistics c(•) on the set CS(w) such that G(β−1) w (x1 = 1, x2 = 1, . . .) = ∑ a∈CS(w) βc(a). (3) Let w be a vexillary permutation. Find a determinantal formula for the β-Grothendieck polynomial G (β) w (X). (4) Let w be a permutation. Find a geometric interpretation of coefficients of the polynomials S (β) w (xi = 1) and S (β) w (xi = q, xj = 1, ∀ j 6= i). For example, let w ∈ Sn be an involution, i.e., w2 = 1, and w′ ∈ Sn+1 be the image of w under the natural embedding Sn ↪→ Sn+1 given by w ∈ Sn −→ (w, n+ 1) ∈ Sn+1. It is well-known, see, e.g., [77, 142], that the multiplicity me,w of the 0-dimensional Schubert cell {pt} = Y w (n+1) 0 in the Schubert variety Y w′ is equal to the specialization Sw(xi = 1) of the Schubert polynomial Sw(Xn). Therefore one can consider the polynomial S (β) w (xi = 1) as a β-deformation of the multiplicity me,w. Question 5.39. What is a geometrical meaning of the coefficients of the polynomial S (β) w (xi = 1) ∈ N[β]? Conjecture 5.40. The polynomial S (β) w (xi = 1) is a unimodal polynomial for any permuta- tion w. 5.2.4 Specialization of Schubert polynomials Let n, k, r be positive integers and p, b be non-negative integers such that r ≤ p + 1. It is well-known [92] that in this case there exists a unique vexillary permutation $ := $λ,φ ∈ S∞ which has the shape λ = (λ1, . . . , λn+1) and the flag φ = (φ1, . . . , φn+1), where λi = (n− i+ 1)p+ b, φi = k + 1 + r(i− 1), 1 ≤ i ≤ n+ 1− δb,0. According to a theorem by M. Wachs [139], the Schubert polynomial S$(X) admits the following determinantal representation S$(X) = DET ( hλi−i+j(Xφi) ) 1≤i,j≤n+1 . Therefore we have S$(1) := S$(x1 = 1, x2 = 1, . . .) = DET (( (n− i+ 1)p+ b− i+ j + k + (i− 1)r k + (i− 1)r )) 1≤i,j≤n+1 . We denote the above determinant by D(n, k, r, b, p). On Some Quadratic Algebras 121 Theorem 5.41. D(n, k, r, b, p) = ∏ (i,j)∈An,k,r i+ b+ jp i ∏ (i,j)∈Bn,k,r (k − i+ 1)(p+ 1) + (i+ j − 1)r + r(b+ np) k − i+ 1 + (i+ j − 1)r , where An,k,r = { (i, j) ∈ Z2 ≥0 \| j ≤ n, j < i ≤ k + (r − 1)(n− j) } , Bn,k,r = { (i, j) ∈ Z2 ≥1 \| i+ j ≤ n+ 1, i 6= k + 1 + rs, s ∈ Z≥0 } . It is convenient to re-write the above formula for D(n, k, r, b, p) in the following form D(n, k, r, b, p) = n+1∏ j=1 ((n− j + 1)p+ b+ k + (j − 1)(r − 1))!(n− j + 1)! (k + (j − 1)r)!((n− j + 1)(p+ 1) + b)! × ∏ 1≤i≤j≤n ((k − i+ 1)(p+ 1) + jr + (np+ b)r). Corollary 5.42 (some special cases). (A) The case r = 1. We consider below some special cases of Theorem 5.41 in the case r = 1. To simplify no- tation, we set D(n, k, b, p) := D(n, k, r = 1, b, p). Then we can rewrite the above formula for D(n, k, r, b, p) as follows D(n, k, b, p) = n+1∏ j=1 ((n+ k − j + 1)(p+ 1) + b)!((n− j + 1)p+ b+ k)!(j − 1)! ((n− j + 1)(p+ 1) + b)!((k + n− j + 1)p+ b+ k)!(k + j − 1)! . (1) If k ≤ n+ 1, then D(n, k, b, p) = k∏ j=1 ( (n+ k + 1− j)(p+ 1) + b n− j + 1 )( (k − j)p+ b+ k j ) j!(k − j)!(n− j + 1)! (n+ k − j + 1)! . In particular, • if k = 1, then D(n, 1, b, p) = 1 + b 1 + b+ (n+ 1)p ( (p+ 1)(n+ 1) + b n+ 1 ) := F (p+1) n+1 (b), where F pn(b) := 1+b 1+b+(p−1)n ( pn+b n ) denotes the generalized Fuss–Catalan number, • if k = 2, then D(n, 2, b, p) = (2 + b)(2 + b+ p) (1 + b)(2 + b+ (n+ 1)p)(2 + b+ (n+ 2)p) F (p+1) n+1 (b)F (p+1) n+2 (b), in particular, D(n, 2, 0, 1) = 6 (n+ 3)(n+ 4) Catn+1Catn+2. See [131, A005700] for several combinatorial interpretations of these numbers. 122 A.N. Kirillov (2) Consider the Young diagram (see R.A. Proctor [122]) λ := λn,p,b = { (i, j) ∈ Z≥1 × Z≥1 \| 1 ≤ i ≤ n+ 1, 1 ≤ j ≤ (n+ 1− i)p+ b}. For each box (i, j) ∈ λ define the numbers c(i, j) := n+ 1− i+ j, and l(i,j)(k) =  k + c(p, j) c(i, j) if j ≤ (n+ 1− i)(p− 1) + b, (p+ 1)k + c(i, j) c(i, j) if (n+ 1− i)(p− 1) < j − b ≤ (n+ 1− i)p. Then D(n, k, b, p) = ∏ (i,j)∈λ l(i,j)(k). (5.10) Therefore, D(n, k, b, p) is a polynomial in k with rational coefficients. (3) If p = 0, then D(n, k, b, 0) = dimV gl(b+k) (n+1)k = n+k∏ j=1 ( j + b j )min(j,n+k+1−j) , where for any partition µ, `(µ) ≤ m, V gl(m) µ denotes the irreducible gl(m)-module with the highest weight µ. In particular, D(n, 2, b, 0) = 1 n+ 2 + b ( n+ 2 + b b )( n+ 2 + b b+ 1 ) is equal to the Narayana number N(n+ b+ 2, b), D(1, k, b, 0) = (b+ k)!(b+ k + 1)! k!b!(k + 1)!(b+ 1)! := N(b+ k + 1, k), and therefore the number D(1, k, b, 0) counts the number of pairs of non-crossing lattice paths inside a rectangular of size (b+1)×(k+1), which go from the point (1, 0) (resp. from that (0, 1)) to the point (b + 1, k) (resp. to that (b, k + 1)), consisting of steps U = (1, 0) and R = (0, 1), see [131, A001263], for some list of combinatorial interpretations of the Narayana numbers. (4) If p = b = 1, then D(n, k, 1, 1) = C (k) n+k+1 := ∏ 1≤i≤j≤n+1 2k + i+ j i+ j . (5) If p = 1 and b is odd integer, then D(n, k, b, 1) is equal to the dimension of the irre- ducible representation of the symplectic Lie algebra Sp(b+ 2n+ 1) with the highest weight kωn+1 (R.A. Proctor [120, 121]). (6) If p = 1 and b = 0, then D(n, k, 1, 0) = D(n− 1, k, 1, 1) = ∏ 1≤i≤j≤n 2k + i+ j i+ j = C (k) n+k, see section on Grothendieck and Narayana polynomials. (7) Let $λ be a unique dominant permutation of shape λ := λn,p,b and ` := `n,p,b = 1 2(n + 1)(np+ 2b) be its length (cf. [44]). Then∑ a∈R($λ) ∏̀ i=1 (x+ ai) = `!B(n, x, p, b). Here for any permutation w of length l, we denote by R(w) the set {a = (a1, . . . , al)} of all reduced decompositions of w. On Some Quadratic Algebras 123 Exercises 5.43. Show that DET ∣∣F (2) n+i+j−2(0) ∣∣ 1≤i,j≤k = k∏ j=1 F (2) n+j−1(0) ( k+1 2 ) !∏ 1≤i≤k−1 1≤j≤k (n+ i+ j) , D(n, k, b, 1) = k∏ j=1 F (2) n+j(b) ∏ 1≤i≤j≤k (b+ i+ j − 1)∏ 1≤i≤k−1 1≤j≤k (n+ b+ i+ j + 1) . Clearly that if b = 0, then F (2) n (0) = Cn, and D(n, k, 0, 1) is equal to the Catalan–Hankel determinant C (k) n . Finally we recall that the generalized Fuss–Catalan number F (p+1) n+1 (b) counts the number of lattice paths from (0, 0) to (b+ np, n) that do not go above the line x = py, see, e.g., [81]. Comments 5.44. It is well-known, see, e.g., [122] or [134, Vol. 2, Exercise 7.101.b], that the number D(n, k, b, p) is equal to the total number ppλn,p,b(k) of plane partitions55 bounded by k and contained in the shape λn,b,p. More generally, see, e.g., [44], for any partition λ denote by wλ ∈ S∞ a unique dominant permutation of shape λ, that is a unique permutation with the code c(w) = λ. Now for any non-negative integer k consider the so-called shifted dominant permutation w (k) λ which has the shape λ and the flag φ = (φi = k + i− 1, i = 1, . . . , `(λ)). Then S w (k) λ (1) = ppλ(≤ k), where ppλ(≤ k) denotes the number of all plane partitions bounded by k and contained in λ. Moreover,∑ π∈PPλ(≤k) q\|π\| = qn(λ)S w (k) λ ( 1, q−1, q−2, . . . ) , where PP λ(≤ k) denotes the set of all plane partitions bounded by k and contained in λ. Exercises 5.45. (1) Show that lim k→∞ S w (k) λ ( 1, q, q2, . . . ) = qn(λ) Hλ(q) , where Hλ(q) = ∏ x∈λ (1− qh(x)) denotes the hook polynomial corresponding to a given partition λ. (2) Let λ = ((n+ `)`, `n) be a fat hook. Show that lim k→∞ qn(λ)S w (k) λ ( 1, q−1, q−2, . . . ) = qs(`,n) Kλ(q) M`(2n+ 2`− 1; q) , where a(`, n) is a certain integer we don’t need to specify in what follows, M`(N ; q) = N∏ j=1 ( 1 1− qj )min(j,N+1−j,`) 55Let λ be a partition. A plane (ordinary) partition bounded by d and shape λ is a filling of the shape λ by the numbers from the set {0, 1, . . . , d} in such a way that the numbers along columns and rows are weakly decreasing. A reverse plane partition bounded by d and shape λ is a filling of the shape λ by the numbers from the set {0, 1, . . . , d} in such a way that the numbers along columns and rows are weakly increasing. 124 A.N. Kirillov denotes the MacMahon generating function for the number of plane partitions fit inside the box N ×N × `, Kλ(q) is a polynomial in q such that Kλ(0) = 1. (a) Show that (1− q)\|λ\| Kλ(q) M`(2n+ 2`− 1; q) ∣∣∣∣ q=1 = 1∏ x∈λ h(x) . (b) Show that Kλ(q) ∈ N[q] and Kλ(1) = M(n, n, `), where M(a, b, c) denotes the number of plane partitions fit inside the box a × b × c. It is well-known, see, e.g., [93, p. 81], that M(a, b, c) = ∏ 1≤i≤a 1≤j≤b 1≤k≤c i+ j + k − 1 i+ j + k − 2 = c∏ i=1 (a+ b+ i− 1)!(i− 1)! (a+ i− 1)!(b+ 1− 1)! = dimV glb+c (ac) . Show that Kλ(q) = ∑ π∈Bn,n,` qwt`(π), where the sum runs over the set of plane partitions π = (πij)1≤i,j≤n fit inside the box Bn,n,` := n× n× `, and wt`(π) = ∑ i,j πij + ` ∑ i πii. (c) Assume as before that λ := ((n+ `)`, `n). Show that lim n→∞ Kλ(q) = M`(q) ∑ µ `(µ)≤` q\|µ\| ( qn(µ)∏ x∈µ(1− qh(x)) )2 , where the sum runs over the set of partitions µ with the number of parts at most `, and n(µ) = ∑ i(i− 1)µi, M`(q) := ∏ j≥1 ( 1− qj )min(j,`) . Therefore the generating function PP (`,0)(q) := ∑ π∈PP (`,0) q\|π\| is equal to ∑ µ `(µ)≤` q\|µ\|  qn(µ)∏ x∈µ (1− qh(x))  2 , where PP (`,k) := {π = (πij)i,j≥1 \|πij ≥ 0, π`+1,`+1 ≤ k}, \|π\| = ∑ i,j πij . (d) Show that PP (`,0)(q) = 1 M`(q)2 ∑ µ, `(µ)≤` (−q)\|µ\|qn(µ)+n(µ′) ( dimq V gl(`) µ )2 , On Some Quadratic Algebras 125 where µ′ denotes the conjugate partition of µ, therefore n(µ′) = ∑ i≥1 ( µi 2 ) . The formula (5.10) is the special case n = m of [109, Theorem 1.2]. In particular, if ` = 1 then one come to following identity 1 (q; q)2 ∞ ∑ k≥0 (−1)kq( k+1 2 ) = ∑ k≥0 qk ( 1 (q; q)k )2 . (e) Let k ≥ 0, ` ≥ 1 be integers. Show that the (fermionic) generating function for the number of plane partitions π = (πij) ∈ PP (`,k) is equal to ∑ π∈PP (`,k) q\|π\| = ∑ µ µ`+1≤k q\|µ\|  qn(µ)∏ x∈µ (1− qh(x))  2 . (B) The case k = 0. (1) D(n, 0, 1, p, b) = 1 for all nonnegative n, p, b. (2) D(n, 0, 2, 2, 2) = VSASM(n), i.e., the number of alternating sign (2n+1)×(2n+1) matrices symmetric about the vertical axis, see, e.g., [131, A005156]. (3) D(n, 0, 2, 1, 2) = CSTCPP(n), i.e., the number of cyclically symmetric transpose comple- ment plane partitions, see, e.g., [131, A051255]. Theorem 5.46. Let $n,k,p be a unique vexillary permutation of the shape λn.p := (n, n − 1, . . . , 2, 1)p and flag φn,k := (k + 1, k + 2, . . . , k + n− 1, k + n). Then G(β−1) $n,1,p(1) = n+1∑ j=1 1 n+ 1 ( n+ 1 j )( (n+ 1)p j − 1 ) βj−1. If k ≥ 2, then Gn,k,p(β) := G (β−1) $n,k,p(1) is a polynomial of degree nk in β, and Coeff [βnk](Gn,k,p(β)) = D(n, k, 1, p− 1, 0). The polynomial n∑ j=1 1 n ( n j )( pn j − 1 ) tj−1 := FNn(t) is known as the Fuss–Narayana polynomial and can be considered as a t-deformation of the Fuss–Catalan number FCp n(0). Recall that the number 1 n ( n j )( pn j−1 ) counts paths from (0, 0) to (np, 0) in the first quadrant, consisting of steps U = (1, 1) and D = (1,−p) and have j peaks (i.e., UD’s), cf. [131, A108767]. For example, take n = 3, k = 2, p = 3, r = 1, b = 0. Then $3,2,3 = [1, 2, 12, 9, 6, 3, 4, 5, 7, 8, 10, 11] ∈ S12, G3,2,3(β) = (1, 18, 171, 747, 1767, 1995, 1001). Therefore, G3,2,3(1) = 5700 = D(3, 2, 3, 0) and Coeff [β6](G3,2,3(β)) = 1001 = D(3, 2, 2, 0). 126 A.N. Kirillov Proposition 5.47 ([110]). The value of the Fuss–Catalan polynomial at t = 2, that is the number n∑ j=1 1 n ( n j )( pn j − 1 ) 2j−1 is equal to the number of hyperplactic classes of p-parking functions of length n, see [110] for definition of p-parking functions, its properties and connections with some combinatorial Hopf algebras. Therefore, the value of the Grothendieck polynomial G (β=1) $n,1,p(1) at β = 1 and xi = 1, ∀ i, is equal to the number of p-parking functions of length n + 1. It is an open problem to find combinatorial interpretations of the polynomials G (β) $n,k,p(1) in the case k ≥ 2. Note finally, that in the case p = 2, k = 1 the values of the Fuss–Catalan polynomials at t = 2 one can find in [131, A034015]. Comments 5.48. (=⇒) The case r = 0. It follows from Theorem 5.32 that in the case r = 0 and k ≥ n, one has D(n, k, 0, p, b) = dimV gl(k+1) λn,p,b = (1 + p)( n+1 2 ) n+1∏ j=1 ((n−j+1)p+b+k−j+1 k−j+1 )( (n−j+1)(p+1)+b n−j+1 ) . Now consider the conjugate ν := νn,p,b := ((n + 1)b, np, (n − 1)p, . . . , 1p) of the partition λn,p,b, and a rectangular shape partition ψ = (k, . . . , k︸︷︷︸ np+b ). If k ≥ np + b, then there exists a unique grassmannian permutation σ := σn,k,p,b of the shape ν and the flag ψ [92]. It is easy to see from the above formula for D(n, k, 0, p, b), that Sσn,k,p,b(1) = dimV gl(k−1) νn,p,b = (1 + p)( n 2) ( k + n− 1 b ) n∏ j=1 (p+ 1)(n− j + 1) (n− j + 1)(p+ 1) + b n∏ j=1 ( k+j−2 (n−j+1)p+b )( (n−j+1)(p+1)+b−1 n−j ) . After the substitution k := np+ b+ 1 in the above formula we will have Sσn,np+b+1,p,b (1) = (1 + p)( n 2) n∏ j=1 (np+b+j−1 (n−j+1)p )( j(p+1)−1 j−1 ) . In the case b = 0 some simplifications are happened, namely, Sσn,k,p,0(1) = (1 + p)( n 2) n∏ j=1 ( k+j−2 (n−j+1)p )( (n−j+1)p+n−j n−j ) . Finally we observe that if k = np+ 1, then n∏ j=1 ( np+j−1 (n−j+1)p )( (n−j+1)p+n−j n−j ) = n∏ j=2 ( np+j−1 (p+1)(j−1) )( j(p+1)−1 j−1 ) = n−1∏ j=1 j!(n(p+ 1)− j − 1)! ((n− j)(p+ 1))!((n− j)(p+ 1)− 1)! := A(p) n , where the numbers A (p) n are integers that generalize the numbers of alternating sign matrices (ASM) of size n× n, recovered in the case p = 2, see [33, 111] for details. On Some Quadratic Algebras 127 Examples 5.49. (1) Let us consider polynomials Gn(β) := G (β−1) σn,2n,2,0(1). If n = 2, then σ2,4,2,0 = 235614 ∈ S6, G2(β) = (1, 2,3) := 1 + 2β + 3β2. Moreover, Rσ2,4,2,0(q;β) = (1,2)β + 3qβ2. If n = 3, then σ3,6,2,0 = 235689147 ∈ S9, G3(β) = (1, 6, 21, 36, 51, 48,26). Moreover, Rσ3,6,2,0(q;β) = (1, 6, 11, 16,11)β + qβ2(10, 20, 35, 34)β + q2β4(5, 14,26)β, Rσ3,6,2,0(q; 1) = (45, 99, 45)q. If n = 4, then σ4,8,2,0 = [2, 3, 5, 6, 8, 9, 11, 12, 1, 4, 7, 10] ∈ S12, G4(β) = (1, 12, 78, 308, 903, 2016, 3528, 4944, 5886, 5696, 4320, 2280,646). Moreover, Rσ4,8,2,0(q;β) = (1, 12, 57, 182, 392, 602, 763, 730, 493,170)β + qβ2(21, 126, 476, 1190, 1925, 2626, 2713, 2026, 804)β + q2β4(35, 224, 833, 1534, 2446, 2974, 2607, 1254)β + q3β6(7, 54, 234, 526, 909, 1026,646)β, Rσ4,8,2,0(q; 1) = (3402, 11907, 11907, 3402)q = 1701 (2, 7, 7, 2)q. • If n = 5, then σ5,10,2 = [2, 3, 5, 6, 8, 9, 11, 12, 14, 15, 1, 4, 7, 10, 13] ∈ S15, G5(β) = (1, 20, 210, 1420, 7085, 27636, 87430, 230240, 516375, 997790, 1676587, 2466840, 3204065, 3695650, 3778095, 3371612, 2569795, 1610910, 782175, 262200,45885). Moreover, Rσ5,10,2,0(q;β) = (1, 20, 174, 988, 4025, 12516, 31402, 64760, 111510, 162170, 202957, 220200, 202493, 153106, 89355, 35972,7429)β + qβ2(36, 432, 2934, 13608, 45990, 123516, 269703, 487908, 738927, 956430, 1076265, 1028808, 813177, 499374, 213597, 47538)β + q2β4(126, 1512, 9954, 40860, 127359, 314172, 627831, 1029726, 1421253, 1711728, 1753893, 1492974, 991809, 461322, 112860)β + q3β6(84, 1104, 7794, 33408, 105840, 255492, 486324, 753984, 1019538, 1169520, 1112340, 825930, 428895, 117990)β + q4β8(9, 132, 1032, 4992, 17730, 48024, 102132, 173772, 244620, 276120, 128 A.N. Kirillov 240420, 144210,45885)β, Rσ5,10,2,0(q; 1) = (1299078, 6318243, 10097379, 6318243, 1299078)q = 59049(22, 107, 171, 107, 22)q. We are reminded that over the paper we have used the notation (a0, a1, . . . , ar)β := r∑ j=0 ajβ j , etc. One can show that deg[β] Gn(β) = n(n− 1), deg[q] Rσn,2n,2,0(q, 1) = n− 1, and looking on the numbers 3, 26, 646, 45885 we made Conjecture 5.50. Let a(n) := Coeff[βn(n−1)] (Gn(β)). Then a(n) = VSASM(n) = OSASM(n) = n−1∏ j=1 (3j + 2)(6j + 3)!(2j + 1)! (4j + 2)!(4j + 3)! , where VSASM(n) is the number of alternating sign (2n+1)× (2n+1) matrices symmetric about the vertical axis, OSASM(n) is the number of 2n × 2n off-diagonal symmetric alternating sign matrices. See [131, A005156], [111] and references therein, for details. Conjecture 5.51. Polynomial Rσn,2n,2,0(q; 1) is symmetric and Rσn,2n,2,0(0; 1) = A20342(2n− 1), see [131]. (2) Let us consider polynomials Fn(β) := G (β−1) σn,2n+1,2,0(1). If n = 1, then σ1,3,2,0 = 1342 ∈ S4, F2(β) = (1,2) := 1 + 2β. If n = 2, then σ2,5,2,0 = 1346725 ∈ S7, F3(β) = (1, 6, 11, 16,11). Moreover, Rσ2,5,2,0(q;β) = (1, 2,3)β + qβ(4, 8, 12)β + q2β3(4,11)β. If n = 3, then σ3,7,2,0 = [1, 3, 4, 6, 7, 9, 10, 2, 5, 8] ∈ S10, F4(β) = (1, 12, 57, 182, 392, 602, 763, 730, 493,170). Moreover, Rσ3,7,2,0(q;β) = (1, 6, 21, 36, 51, 48,26)β + qβ(6, 36, 126, 216, 306, 288, 156)β + q2β3(20, 125, 242, 403, 460, 289)β + q3β5(6, 46, 114, 204,170)β, Rσ3,7,2,0(q; 1) = (189, 1134, 1539, 540)q = 27(7, 42, 57, 20)q. On Some Quadratic Algebras 129 If n = 4, then σ4,9,2,0 = [1, 3, 4, 6, 7, 9, 10, 12, 13, 2, 5, 8, 11] ∈ S13, F5(β) = (1, 20, 174, 988, 4025, 12516, 31402, 64760, 111510, 162170, 202957, 220200, 202493, 153106, 89355, 35972,7429). Moreover, Rσ4,9,2,0(q;β) = (1, 12, 78, 308, 903, 2016, 3528, 4944, 5886, 5696, 4320, 2280,646)β + qβ(8, 96, 624, 2464, 7224, 16128, 28224, 39552, 47088, 45568, 34560, 18240, 5168)β + q2β3(56, 658, 3220, 11018, 27848, 53135, 78902, 100109, 103436, 84201, 47830, 14467)β + q3β5(56, 728, 3736, 12820, 29788, 50236, 72652, 85444, 78868, 50876, 17204)β + q4β7(8, 117, 696, 2724, 7272, 13962, 21240, 24012, 18768,7429)β, Rσ4,9,2,0(q; 1) = (30618, 244944, 524880, 402408, 96228)q = 4374(7, 56, 120, 92, 22)q. One can show that Fn(β) is a polynomial in β of degree n2, and looking on the numbers 2, 11, 170, 7429 we made Conjecture 5.52. Let b(n) := Coeff [β(n−1)2 ] (Fn(β)). Then b(n) = CSTCPP(n). In other words, b(n) is equal to the number of cyclically symmetric transpose complement plane partitions in an 2n× 2n× 2n box. This number is known to be n−1∏ j (3j + 1)(6j)!(2j)! (4j + 1)!(4j)! , see [131, A051255], [18, p. 199]. It ease to see that polynomial Rσn,2n+1,2,0(q; 1) has degree n. Conjecture 5.53. Coeff [βn] ( Rσn,2n+1,2,0(q; 1) ) = A20342(2n), see [131]; Rσn,2n+1,2,0(0; 1) = A (1) QT(4n; 3) = 3n(n−1)/2ASM(n), see [83, Theorem 5] or [131, A059491]. Proposition 5.54. One has Rσ4,2n+1,2,0(0;β) = Gn(β) = G(β−1) σn,2n,2,0(1), Rσn,2n,2,0(0, β) = Fn(β) = G(β−1) σn,2n+1,2,0 (1). Finally we define (β, q)-deformations of the numbers VSASM(n) and CSCTPP(n). To ac- complish these ends, let us consider permutations w−k = (2, 4, . . . , 2k, 2k − 1, 2k − 3, . . . , 3, 1), w+ k = (2, 4, . . . , 2k, 2k + 1, 2k − 1, . . . , 3, 1). 130 A.N. Kirillov Proposition 5.55. One has Sw−k (1) = VSAM(k), Sw+ k (1) = CSTCPP(k). Therefore the polynomials G (β−1) w−k (x1 = q, xj = 1, ∀ j ≥ 2) and G (β−1) w+ k (x1 = q, xj = 1, ∀ j ≥ 2) define (β, q)-deformations of the numbers VSAM(k) and CSTCPP(k) respectively. Note that the inverse permutations (w−k )−1 = (2k, 1︸︷︷︸, . . . , 2k + 1− i, i︸︷︷︸, . . . , k + 1, k︸︷︷︸), (w+ k )−1 = (2k + 1, 1︸︷︷︸, . . . , 2k + 2− j, j︸︷︷︸, . . . , k + 2, k︸︷︷︸, k + 1) also define a (β, q)-deformation of the numbers considered above. Problem 5.56. It is well-known, see, e.g., [37, p. 43], that the set VSASM(n) of alternating sign (2n+1)×(2n+1) matrices symmetric about the vertical axis has the same cardinality as the set SYT2(λ(n),≤ n) of semistandard Young tableaux of the shape λ(n) := (2n−1, 2n−3, . . . , 3, 1) filled by the numbers from the set {1, 2, . . . , n}, and such that the entries are weakly increasing down the anti-diagonals. On the other hand, consider the set CS(w−k ) of compatible sequences, see, e.g., [13, 42], corresponding to the permutation w−k ∈ S2k. Challenge 5.57. Construct bijections between the sets CS(w−k ), SYT2(λ(k),≤ k) and VSASM(k). Remark 5.58. One can compute the principal specialization of the Schubert polynomial cor- responding to the transposition tk,n := (k, n− k) ∈ Sn that interchanges k and n− k, and fixes all other elements of [1, n]. Proposition 5.59. q(n−1)(k−1)Stk,n−k ( 1, q−1, q−2, q−3, . . . ) = k∑ j=1 (−1)j−1q( j 2) [ n− 1 k − j ] q [ n− 2 + j k + j − 1 ] q = n−2∑ j=1 qj ([ j + k − 2 k − 1 ] q )2 . Exercises 5.60. (1) Show that if k ≥ 1, then Coeff [qkβ2k](Rσn,2n,2,0(q; t)) = ( 2n− 1 2k ) , Coeff [qkβ2k−1](Rσn,2n+1,2,0(q; t)) = ( 2n 2k − 1 ) . (2) Let n ≥ 1 be a positive integer, consider “zig-zag” permutation w = ( 1 2 3 4 . . . 2k + 1 2k + 2 . . . 2n− 1 2n 2 1 4 3 . . . 2k + 2 2k + 1 . . . 2n 2n− 1 ) ∈ S2n. Show that Rw(q, β) = n−1∏ k=0 ( 1− β2k 1− β + qβ2k ) . On Some Quadratic Algebras 131 (3) Let σk,n,m be grassmannian permutation with shape λ = (nm) and flag φ = (k+1)m, i.e., σk,n,m = ( 1 2 . . . k k + 1 . . . k + n k + n+ 1 . . . k + n+m 1 2 . . . k k +m+ 1 . . . k +m+ n k + 1 . . . k +m ) . Clearly σk+1,n,m = 1× σk,n,m. Show that the coefficient Coeffβm(Rσk,n,m(1, β)) is equal to the Narayana number N(k+n+ m, k). (4) Consider permutation w := w(n) = (w1, . . . , w2n+1), where w2k−1 = 2k+1 for k = 1, . . . , n, w2n+1 = 2n, w2 = 1 and w2k = 2k − 2 for k = 2, . . . , n. For example, w(3) = (3152746). We set w(0) = 1. Show that the polynomial S (β) w (xi = 1, ∀i) has degree n(n− 1) and the coefficient Coeffβn(n−1)(S (β) w (xi = 1, ∀ i)) is equal to the n-th Catalan number Cn. Note that the specialization S (β) w (xi = 1)\|β=1 is equal to the 2n-th Euler (or up/down) number, see [131, A000111]. More generally, consider permutation w (n) k := 1k × w(n) ∈ Sk+2n+1, and polynomials Pk(z) = ∑ j≥0 (−1)jS w (j) k−2j (xi = 1)zk−2j , k ≥ 0. Show that∑ k≥0 Pk(z) tk k! = exp(tz) sech(t). The polynomials Pk(z) are well-known as Swiss–Knife polynomials, see [131, A153641], where one can find an overview of some properties of the Swiss–Knife polynomials. (5) Assume that n = 2k + 3, k ≥ 1, and consider permutation vn = (v1, . . . , vn) ∈ Sn, where v2a+1 = 2a + 3, a = 0, . . . , n − 1, w2 = 1 and w2a = 2a − 2, a = 2, . . . , k + 1. For example, v4 = [31527496, 11, 8, 10] and Sv4(1) = 50521 = E10. Show that Svn(q, xi = 1, ∀ i ≥ 2) = (n− 2)En−3q 2 + · · ·+ 2k−1(k − 1)!qk+2, Svn(xi = 1, ∀ i ≥ 1) = En−1. Set β = d− 1, consider polynomials En(q, d) = G (β) vn (x1 = q, xi = 1, ∀ i ≥ 2). Clearly, see the latter formula, En(1, 1) = En−1. Give a combinatorial prove that En(q, d) ∈ N[q, d], that is to give combinatorial interpretation(s) of coefficients of the polynomial En(q, d). Show that degd En(1, d) = n(n+ 1) and the leading coefficient is equal to the Catalan num- ber Cn+1. (6) Consider permutation u := un = (u1, . . . , u2n) ∈ S2n, n ≥ 2, where u1 = 2, u2k+1 = 2k−1, k = 1, . . . , n, u2k = 2k + 2, k = 1, . . . , n− 1, u2n = 2n− 1. For example, u4 = (24163857). Now consider polynomial R(k) n (q) = S1k×un(x1 = q, xi = 1, ∀ i ≥ 2). Show that R (k) n (1) = ( 2n+k−1 k ) E2n−1, where E2k−1, k ≥ 1, denotes the Euler number, see [131, A00111]. In particular, R (1) n (1) = 22n−1Gn, where Gn denotes the unsigned Genocchi number, see [131, A110501]. Show that degq R (k) n (q) = n and Coeffqn ( R (0) n (q) ) = (2n− 3)!!. (7) Consider permutation wn ∈ S2n+2, where w2 = 1, w4 = 2, and w2k−1 = 2k + 2, 1 ≤ k ≤ n, w2k = 2k − 3, 3 ≤ k ≤ n, 132 A.N. Kirillov w2n+1 = 2n− 3, w2n+2 = 2n− 1. For example, w5 = [4, 1, 6, 2, 8, 3, 10, 5, 12, 7, 9, 11]. Show that Swn(xi = 1, ∀ i) = (2n+ 1)!! ( 22n − 2 ) \|B2n\|, where B2n denotes the Bernoulli numbers56. (8) Consider permutation wk := (2k + 1, 2k − 1, . . . , 3, 1, 2k, 2k − 2, . . . , 4, 2) ∈ S2k+1. Show that S(β−1) wk (x1 = q, xj = 1, ∀ j ≥ 2) = q2k(1 + β)( n 2). (9) Consider permutations σ+ k = (1, 3, 5, . . . , 2k+ 1, 2k+ 2, 2k, . . . , 4, 2) and σ−k = (1, 3, 5, . . ., 2k + 1, 2k, 2k − 2, . . . , 4, 2), and define polynomials S±k (q) = Sσ±k (x1 = q, xj = 1, ∀ j ≥ 2). Show that S+ k (0) = VSASM(k), S+ k (1) = VSASM(k + 1), ∂ ∂q S+ k (q)\|q=0 = 2kS+ k (0), Coeffqk(S+ k (q)) = CSTCPP(k + 1), S−k (0) = CSTCPP(k), S−k (1) = CSTCPP(k + 1), ∂ ∂q S−k (q)\|q=0 = (2k − 1)S−k (0), Coeffqk(S−k (q)) = VSASM(k). Let’s observe that σ±k = 1 × τ±k−1, where τ+ k = (2, 4, . . . , 2k, 2k + 1, 2k − 1, . . . , 3, 1) and τ−k = (2, 4, . . . , 2k, 2k − 1, 2k − 3, . . . , 3, 1). Therefore, Sτ±k (x1 = q, xj = 1, ∀j ≥ 2) = qS±k−1(q). Recall that CSTCPP(n) denotes the number of cyclically symmetric transpose compliment plane partitions in a 2n × 2n box, see, e.g., [131, A051255], and VSASM(n) denotes the number of alternating sign (2n+1)×(2n+1) matrices symmetric the vertical axis, see, e.g., [131, A005156]. It might be well to point out that Sσ+ n−1 (x1 = x, xi = 1, ∀ i ≥ 2) = G2n−1,n−1(x, y = 1), Sσ−n (x1 = x, xi = 1, ∀ i ≥ 2) = F2n,n−1(x, y = 1), where (homogeneous) polynomials Gm,n(x, y) and Fm,n(x, y) are defined in [123], and related with integral solutions to Pascal’s hexagon relations fm−1,nfm+1,n + fm,n−1fm,n+1 = fm−1,n−1fm+1,n+1, (m,n) ∈ Z2. (10) Consider permutation un = ( 1 2 . . . n n+ 1 n+ 2 n+ 3 . . . 2n 2 4 . . . 2n 1 3 5 . . . 2n− 1 ) , 56See, e.g., https://en.wikipedia.org/wiki/Bernoulli_number. https://en.wikipedia.org/wiki/Bernoulli_number On Some Quadratic Algebras 133 and set u (k) n := 12k+1 × un. Show that G (β−1) u (k) n (xi = 1, ∀ i ≥ 1) = (1 + β)( n+1 2 )G ((β)2−1) 1k×w(n+1) 0 (xi = 1, ∀ i ≥ 1), where w (n+) 0 denotes the permutation (n+ 1, n, n− 1, . . . , 2, 1). (11) Let n ≥ 0 be an integer. Consider permutation un = 1n × 321 ∈ S3+n. Show that Sun(x1 = t, xi = 1, ∀ i ≥ 2) = 1 4 ( 2n+ 2 3 ) + n 2 ( 2n+ 2 1 ) t+ 1 2 ( 2n+ 2 1 ) t2. Consider permutation vn := 1n × 4321 ∈ Sn+4. Show that Svn(x1 = t, xi = 1, ∀ i ≥ 2) = 1 24 ( 2n+ 4 5 )( 2n+ 2 1 ) + 1 2 ( 2n+ 4 5 ) t+ n 4 ( 2n+ 4 3 ) t2 + 1 4 ( 2n+ 4 3 ) t3. (12) Show that ∑ (a,b,c)∈(Z≥0)3 qa+b+c [ a+ b b ] q [ a+ c c ] q [ b+ c b ] q = 1 (q; q)3 ∞ ∑ k≥2 (−1)k ( k 2 ) q( k 2)−1  . It is not difficult to see that the left hand side sum of the above identity counts the weighted number of plane partitions π = (πij) such that πi,j ≥ 0, πij ≥ max(πi+1,j , πi,j+1), πij ≤ 1 if i ≥ 2 and j ≥ 2, and the weight wt(π) := ∑ i,j πij . (13) Let λ = (λ1 ≥ λ2 ≥ · · · ≥ λp > 0) be a partition of size n. For an integer k such that 1 ≤ k ≤ n− p define a grassmannian permutation w (k) λ = [1, . . . , k, λp + k + 1, λp−1 + k + 2, . . . , λ1 + k + p, a1, . . . , an−p−k], where we denote by (a1 < a2 < · · · < an−k−p) the complement [1, n]\(1, . . . , k, λp+k+ 1, λp−1 + k + 2, . . . , λ1 + k + p)]. Show that the Grothendieck polynomial Gλ(β) := Gβ−1 wλk (1n) is a polynomial of β with nonnegative coefficients. Clearly, Gλ(1) = dimV Gl(k+`(λ)) λ . Find a combinatorial interpretations of polynomial Gλ(β). Final remark, it follows from the seventh exercise listed above, that the polynomials S (β) σ±k (x1 = q, xj = 1, ∀ j ≥ 2) define a (q, β)-deformation of the number VSASM(k) (the case σ+ k ) and the number CSTCPP(k) (the case σ−k ), respectively. 5.2.5 Specialization of Grothendieck polynomials Let p, b, n and i, 2i < n be positive integers. Denote by T (i) p,b,n the trapezoid, i.e., a convex quadrangle having vertices at the points (ip, i), (ip, n− i), (b+ ip, i) and (b+ (n− i)p, n− i). 134 A.N. Kirillov Definition 5.61. Denote by FC (i) b,p,n the set of lattice path from the point (ip, i) to that (b + (n − i)p, n − i) with east steps E = (0, 1) and north steps N = (1, 0), which are located inside of the trapezoid T (i) p,b,n. If p ∈ FC (i) b,p,n is a path, we denote by p(p) the number of peaks, i.e., p(p) = NE(p) + Ein(p) +Nend(p), where NE(p) is equal to the number of steps NE resting on path p; Ein(p) is equal to 1, if the path p starts with step E and 0 otherwise; Nend(p) is equal to 1, if the path p ends by the step N and 0 otherwise. Note that the equality Nend(p) = 1 may happened only in the case b = 0. Definition 5.62. Denote by FC (k) b,p,n the set of k-tuples P = (p1, . . . , pk) of non-crossing lattice paths, where for each i = 1, . . . , k, pi ∈ FC (i) b,p,n. Let FC (k) b,p,n(β) := ∑ P∈FC (k) b,p,n βp(P) denotes the generating function of the statistics p(P) := k∑ i=1 p(pi)− k. Theorem 5.63. The following equality holds G(β) σn,k,p,b (x1 = 1, x2 = 1, . . .) = FC (k) p,b,n+k(β + 1), where σn,k,p,b is a unique grassmannian permutation with shape ((n + 1)b, np, (n − 1)p, . . . , 1p) and flag (k, . . . , k)︸︷︷︸ np+b . 5.3 The “longest element” and Chan–Robbins–Yuen polytope57 5.3.1 The Chan–Robbins–Yuen polytope CRYn Assume additionally, cf. [133, Exercise 6.C8(d)], that the condition (a) in Definition 5.1 is replaced by that (a′) xij and xkl commute for all i, j, k and l. Consider the element w (n) 0 := ∏ 1≤i<j≤n xij . Let us bring the element w (n) 0 to the reduced form, that is, let us consecutively apply the defining relations (a′) and (b) to the element w (n) 0 in any order until unable to do so. Denote the resulting polynomial by Qn(xij ;α, β). Note that the polynomial itself depends on the order in which the relations (a′) and (b) are applied. We denote by Qn(β) the specialization xij = 1 for all i and j, of the polynomial Qn(xij ;α = 0, β). 57Some results of this section, e.g., Theorems 5.63 and 5.65, has been proved independently and in greater generality in [102]. On Some Quadratic Algebras 135 Example 5.64. Q3(β) = (2, 1) = 1 + (β + 1), Q4(β) = (10, 13, 4) = 1 + 5(β + 1) + 4(β + 1)2, Q5(β) = (140, 336, 280, 92, 9) = 1 + 16(β + 1) + 58(β + 1)2 + 56(β + 1)3 + 9(β + 1)4, Q6(β) = 1 + 42(β + 1) + 448(β + 1)2 + 1674(β + 1)3 + 2364(β + 1)4 + 1182(β + 1)5 + 169(β + 1)6, Q7(β) = (1, 99, 2569, 25587, 114005, 242415, 248817, 118587, 22924, 1156)β+1, Q8(β) = (1, 219, 12444, 279616, 2990335, 16804401, 52421688, 93221276, 94803125, 53910939, 16163947, 2255749, 108900)β+1. What one can say about the polynomial Qn(β) := Qn(xij ;β)\|xij=1, ∀ i,j? It is known, [133, Exercise 6.C8(d)], that the constant term of the polynomial Qn(β) is equal to the product of Catalan numbers n−1∏ j=1 Cj . It is not difficult to see that if n ≥ 3, then Coeff [β+1](Qn(β)) = 2n − 1 − ( n+1 2 ) , see [131, A002662], for a number of combinatorial interpretations of the numbers 2n − 1− ( n+1 2 ) . Theorem 5.65. One has Qn(β − 1) = ∑ m≥0 ι(CRYn+1,m)βm  (1− β)( n+1 2 )+1, where CRYm denotes the Chan–Robbins–Yuen polytope [20, 21], i.e., the convex polytope given by the following conditions: CRYm = { (aij) ∈ Matm×m(Z≥0) } such that (1) ∑ i aij = 1, ∑ j aij = 1, (2) aij = 0 if j > i+ 1. Here for any integral convex polytope P ⊂ Zd, ι(P, n) denotes the number of integer points in the set nP ∩ Zd. In particular, the polynomial Qn(β) does not depend on the order in which the relations (a′) and (b) have been applied. Now let us denote by Q̂n(q, t;α, β) the specialization xij = 1, i < j < n, and xi,n = q if i = 2, . . . , n− 1, x1,n = t of the (reduced) polynomial Qn(xij ;α, β) obtained by applying the relations (a′) and (b) in a certain order. The polynomial Qn(xij ;α, β) itself depends on the order selected. To define polynomials which are frequently appear in Section 5, we apply the rules (a) and (b) stated in Definition 5.1 to a given monomial xi1,j1 · · ·xip,jp ∈ ÂCYBn(α, β) consequently according to the order in which the monomial taken has been written. We set Qn(t, α, β) := Q̂n(q = t, t;α, β). Conjecture 5.66. Let n ≥ 3 and write Qn(t = 1;α, β) = ∑ k≥0 (1 + β)kck,n(α), 136 A.N. Kirillov then ck,n(α) ∈ Z≥0[α]. The polynomial Qn(t, β, α = 0) has degree dn := [ (n−1)2 4 ] with respect to β. Write Qn(t, β) := Qn(t;α = 0, β) = tn−2 dn∑ k=0 c(k) n (t)βk. Then c (dn) n (1) = a2 n for some non-negative integer an. Moreover, there exists a polynomial an(t) ∈ N[t] such that c(dn) n (t) = an(1)an(t), an(0) = an−1. The all roots of the polynomial Qn(β) belong to the set R<−1. For example, Q4(t = 1;α, β) = (1, 5, 4)β+1 + α(5, 7)β+1 + 3α2, Q5(t = 1;α, β) = (1, 16, 58, 56, 9)β+1 + α(16, 109, 146, 29)β+1 + α2(51, 125, 34)β+1 + α3(35, 17)β+1, c (6) 6 = 13(2, 3, 3, 3, 2), c (9) 7 (t) = 34(3, 5, 6, 6, 6, 5, 3), c (12) 8 (t) = 330(13, 27, 37, 43, 45, 45, 43, 37, 27, 13), Q4(t, β, α = 0)t−1 = t2 + (β + 1) ( 3t+ 2t2 ) + (β + 1)2(t+ 1)2, Q̂4(q, t;α = 0, β) = ( qt2 + t3 + 2qt3 + q2t3 + q3t3 + t4 + 2qt4 + q2t4 ) + ( 2qt2 + 2t3 + 3qt3 + 2q2t3 + 2t4 + 2qt4 ) β + ( t2 + t3 ) (q + t)β2, Q̂5(q, t;α = 0, β) = ( 3q2t+ q3t+ 5qt2 + 6q2t2 + 2q3t2 + 2t3 + 10qt3 + 10q2t3 + 6q3t3 + 3q4t3 + 3q5t3 + 2q6t3 + 3t4 + 11qt4 + 11q2t4 + 8q3t4 + 5q4t4 + 3q5t4 + 3t5 + 9qt5 + 9q2t5 + 6q3t5 + 3q4t5 + 2t6 + 6qt6 + 6q2t6 + 2q3t6 ) + ( 9q2t+ 2q3t+ 17qt2 + 18q2t2 + 4q3t2 + 7t3 + 31qt3 + 29q2t3 + 15q3t3 + 10q4t3 + 7q5t3 + 10t4 + 31qt4 + 29q2t4 + 18q3t4 + 10q4t4 + 10t5 + 24qt5 + 21q2t5 + 10q3t5 + 6t6 + 12qt6 + 6q2t6 ) β + ( 9q2t+ q3t+ 21qt2 + 18q2t2 + 2q3t2 + 9t3 + 34qt3 + 28q2t3 + 14q3t3 + 9q4t3 + 12t4 + 30qt4 + 24q2t4 + 12q3t4 + 12t5 + 21qt5 + 12q2t5 + 6t6 + 6qt6 ) β2 + ( 3q2t+ 11qt2 + 6q2t2 + 5t3 + 15qt3 + 10q2t3 + 5q3t3 + 6t4 + 11qt4 + 6q2t4 + 6t5 + 6qt5 + 2t6 ) β3 + ( 2qt2 + t3 + 2qt3 + q2t3 + t4 + qt4 + t5 ) β4. Note that polynomials Q̂n(q, t;α = 0, β = 0) give rise to a two parameters deformation of the product of Catalan’s numbers C1C2 · · ·Cn−1. Are there combinatorial interpretations of these polynomials and polynomials Q̂n(q, t;α = 0, β)? Comments 5.67. We expect that for each integer n ≥ 2 the set Ψn+1 := w ∈ S2n−1 \|Sw(1) = n∏ j=1 Catj  On Some Quadratic Algebras 137 is non empty, whereas the setw ∈ S2n−2 \|Sw(1) = n∏ j=1 Catj  is empty. For example, Ψ4 = {[1, 5, 3, 4, 2]}, Ψ5 = {[1, 5, 7, 3, 2, 6, 4], [1, 5, 4, 7, 2, 6, 3]}, Ψ6 = { w := [1, 3, 2, 8, 6, 9, 4, 5, 7], w−1, . . . } , Ψ7 = {???}, but one can check that for w = [2358, 10, 549, 12, 11] ∈ S12 Sw(1) = 776160 = 6∏ j=2 Catj . More generally, for any positive integer N define κ(N) = min{n \| ∃w ∈ Sn such that Sw(1) = N}. It is clear that κ(N) ≤ N + 1. Problem 5.68. Compute the following numbers κ(n!), κ  n∏ j=1 Catj  , κ(ASM(n)), κ ( (n+ 1)n−1 ) . For example, 10 ≤ κ(ASM(6) = 7436) ≤ 12. Indeed, take w = [716983254, 10, 12, 11] ∈ S12. One can show that Sw(x1 = t, xi = 1, ∀ i ≥ 2) = 13t6(t+ 10)(15t+ 37), so that Sw(1) = ASM(6); κ(64) = 9, namely, one can take w = [157364298]. Question 5.69. Let N be a positive integer. Does there exist a vexillary (grassmannian?) permutation w ∈ Sn such that n ≤ 2κ(N) and Sw(1) = N? For example, w = [1, 4, 5, 6, 8, 3, 5, 7] ∈ S8 is a grassmannian permutation such that Sw(1) = 140, and Rw(1, β) = (1, 9, 27, 43, 38, 18, 4). Remark 5.70. We expect that for n ≥ 5 there are no permutations w ∈ S∞ such that Qn(β) = S (β) w (1). The numbers Cn := n∏ j=1 Catj appear also as the values of the Kostant partition function of the type An−1 on some special vectors. Namely, Cn = KΦ(1n)(γn), where γn = ( 1, 2, 3, . . . , n− 1,− ( n 2 )) , see, e.g., [133, Exercise 6.C10], and [69, pp. 173–178]. More generally [69, Exercise g, p. 177, (7.25)], one has KΦ(1n)(γn,d) = ppδn(d)Cn−1 = n+d−2∏ j=d 1 2j + 1 ( n+ d+ j 2j ) , 138 A.N. Kirillov where γn,d = (d+ 1, d+ 2, . . . , d+ n− 1,−n(2d+ n− 1)/2), ppδn(d) denotes the set of reversed (weak) plane partitions bounded by d and contained in the shape δn = (n − 1, n − 2, . . . , 1). Clearly, ppδn(1) = ∏ 1≤i<j≤n i+j+1 i+j−1 = Cn, where Cn is the n-th Catalan number58. Conjecture 5.71. For any permutation w ∈ Sn there exists a graph Γw = (V,E), possibly with multiple edges, such that the reduced volume ṽol(FΓw) of the flow polytope FΓw , see, e.g., [132] for a definition of the former, is equal to Sw(1). For a family of vexillary permutations wn,p of the shape λ = pδn+1 and flag φ = (1, 2, . . ., n − 1, n) the corresponding graphs Γn,p have been constructed in [101, Section 6]. In this case the reduced volume of the flow polytope FΓn,p is equal to the Fuss–Catalan number 1 1 + (n+ 1)p ( (n+ 1)(p+ 1) n+ 1 ) = Swn,p(1), cf. Corollary 5.33. Exercises 5.72. (a) Show that the polynomial Rn(t) := t1−nQn(t; 0, 0) is symmetric (unimodal?), and Rn(0) = n−2∏ k=1 Catk. For example, R4(t) = (1 + t) ( 2 + t+ 2t2 ) , R5(t) = 2(5, 10, 13, 14, 13, 10, 5)t, R6(t) = 10(2, 3, 2)t(7, 7, 10, 13, 10, 13, 10, 7, 7)t, R7(t) = 30 ( 196 + · · ·+ 196t15 ) . Note that Rn(1) = n−1∏ k=1 Catk. (b) More generally, write as before, Qn(t; 0, β) = tn−2 ∑ k≥0 c(k) n (t)βk. Show that the polynomials c (k) n (t) are symmetric (unimodal?) for all k and n. (c) Consider a reduced polynomial Rn({xij}) of the element∏ 1≤i<j≤n (i,j)6=(n−1,n) xij ∈ ÂCYB(α = β = 0)ab, see Definition 5.1. Here we assume additionally, that all elements {xij} are mutually commute. Define polynomial R̃n(q, t) to be the following specialization xij −→ 1 if i < j < n− 1, xi,n−1 −→ q, xi,n −→ t, ∀ i of the polynomial Rn({xij}) in question. Show that polynomials R̃n(q, t) are well-defined, and R̃n(q, t) = R̃n(t, q). 58For example, if n = 3, there exist 5 reverse (weak) plane partitions of shape δ3 = (2, 1) bounded by 1, namely reverse plane partitions {( 0 0 0 ) , ( 0 0 1 ) , ( 0 1 0 ) , ( 0 1 1 ) , ( 1 1 1 )} . On Some Quadratic Algebras 139 Examples 5.73. R4(t, β) = (2, 3, 3, 2)t + (4, 5, 4)tβ + (2, 2)tβ 2, R5(t, β) = (10, 20, 26, 28, 26, 20, 10)t + (33, 61, 74, 74, 61, 33)tβ + (39, 65, 72, 65, 39)tβ 2 + (19, 27, 27, 19)tβ 3 + (3, 3, 3)tβ 4, R6(t, β) = (140, 350, 550, 700, 790, 820, 790, 700, 550, 350, 140)t + (686, 1640, 2478, 3044, 3322, 3322, 3044, 2478, 1640, 686)tβ + (1370, 3106, 4480, 5280, 5537, 5280, 4480, 3106, 1370)tβ 2 + (1420, 3017, 4113, 4615, 4615, 4113, 3017, 1420)tβ 3 + (800, 1565, 1987, 2105, 1987, 1565, 800)tβ 4 + (230, 403, 465, 465, 403, 230)tβ 5 + (26, 39, 39, 39, 26)tβ 6, R6(1, β) = (5880, 22340, 34009, 26330, 10809, 2196, 169)β, R7(t, β) = (5880, 17640, 32340, 47040, 59790, 69630, 76230, 79530, 79530, 76230, 69630, 59790, 47040, 32340, 17640, 5880)t + (39980, 116510, 208196, 295954, 368410, 420850, 452226, 462648, 452226, 420850, 368410, 295954, 208196, 116510, 39980)tβ + (118179, 333345, 578812, 802004, 975555, 1090913, 1147982, 1147982, 1090913, 975555, 802004, 578812, 333345, 118179)tβ 2 + (198519, 539551, 906940, 1221060, 1447565, 1580835, 1624550, 1580835, 1447565, 1221060, 906940, 539551, 198519)tβ 3 + (207712, 540840, 875969, 1141589, 1314942, 1398556, 1398556, 1314942, 1141589, 875969, 540840, 207712)tβ 4 + (139320, 344910, 535107, 671897, 749338, 773900, 749338, 671897, 535107, 344910, 139320)tβ 5 + (59235, 137985, 203527, 244815, 263389, 263389, 244815, 203527, 137985, 59235)tβ 6 + (15119, 32635, 45333, 51865, 53691, 51865, 45333, 32635, 15119)tβ 7 + (2034, 3966, 5132, 5532, 5532, 5132, 3966, 2034)β8 + (102, 170, 204, 204, 204, 170, 102)tβ 9, R7(1, β) = (776160, 4266900, 10093580, 13413490, 10959216, 5655044, 1817902, 343595, 33328, 1156)β. 5.3.2 The Chan–Robbins–Mészáros polytope Pn,m Let m ≥ 0 and n ≥ 2 be integers, consider the reduced polynomial Qn,m(t, β) corresponding to the element Mn.m :=  n∏ j=2 x1j m+1 n−2∏ j=2 n∏ k=j+2 xjk. For example, Q2,4(t, β) = (4, 7, 9, 10, 10, 9, 7, 4)t + (10, 17, 21, 22, 21, 17, 10)tβ + (8, 13, 15, 15, 13, 8)tβ 2 + (2, 3, 3, 3, 2)tβ 3, Q2,4(1, β) = (60, 118, 72, 13)β, Q2,5(t, β) = (60, 144, 228, 298, 348, 378, 388, 378, 348, 298, 228, 144, 60)t 140 A.N. Kirillov + (262, 614, 948, 1208, 1378, 1462, 1462, 1378, 1208, 948, 614, 262)tβ + (458, 1042, 1560, 1930, 2142, 2211, 2142, 1930, 1560, 1042, 458)tβ 2 + (405, 887, 1278, 1526, 1640, 1640, 1526, 1278, 887, 405)tβ 4 + (187, 389, 534, 610, 632, 610, 534, 389, 187)tβ 4 + (41, 79, 102, 110, 110, 102, 79, 41)tβ 5 + (3, 5, 6, 6, 6, 5, 3)tβ 6, Q2,5(1, β) = (3300, 11744, 16475, 11472, 4072, 664, 34)β, Q2,6(1, β) = (660660, 3626584, 8574762, 11407812, 9355194, 4866708, 1589799, 310172, 32182, 1320)β, Q2,7(β) = (1, 213, 12145, 279189, 3102220, 18400252, 61726264, 120846096, 139463706, 93866194, 5567810, 7053370, 626730, 16290)β+1. Theorem 5.74. One has Qm,n(1, 1) = n−2∏ k=1 Catk ∏ 1≤i<j≤n−1 2(m+ 1) + i+ j − 1 i+ j − 1 , ∑ k≥0 ι(Pn,m; k)βk = Qm,n(1, β − 1) (1− β)( n+1 2 )+1 , where Pn,m denotes the generalized Chan–Robbins–Yuen polytope defined in [101], and for any integral convex polytope P, ι(P, k) denotes the Ehrhart polynomial of polytope P. Conjecture 5.75. Let n ≥ 3, m ≥ 0 be integers, , write Qm,n(t, β) = ∑ k≥0 c(k) m,n(t)βk, and set b(m,n) := max ( k \| c(k) m,n(t) 6= 0 ) . Denote by c̃m,n(t) the polynomial obtained from that c (b(m,n) m,n (t) by dividing the all coefficients of the latter on their GCD. Then c̃n,m(t) = an+m(t), where the polynomials an(t) := c0,n(t) have been defined in Conjecture 5.66. For example, c2,5(t) = 4a7(t), c2,6(t) = 10a8(t), c3,5(t) = a8(t), c2,7(t) = 10(34, 78, 118, 148, 168, 178, 181, 178, 168, 148, 118, 78, 34) ? = 10a9(t). It is known [69, 99, 100] that n−2∏ k=1 Catk ∏ 1≤i<j≤n−1 2(m+ 1) + i+ j − 1 i+ j − 1 = m+n−2∏ j=m+1 1 2j + 1 ( n+m+ j 2j ) = KAn+1 ( m+ 1,m+ 2, . . . , n+m,−mn− ( n 2 )) . Conjecture 5.76. Let a = (a2, a3, . . . , an) be a sequence of non-negative integers, consider the following element M(a) =  n∏ j=2 x aj 1j  n−1∏ j=2  n∏ k=j+1 xjk  . On Some Quadratic Algebras 141 Let Ra(t1, . . . , tn−1, α, β) be the following specialization xij −→ tj−1 for all 1 ≤ i < j ≤ n of the reduced polynomial Ra(xij) of monomial Ma ∈ ÂCYBn(α, β). Then the polynomial Ra(t1, . . . , tn−1, α, β) is well-defined, i.e., does not depend on an order in which relations (a′) and (b), Definition 5.1, have been applied. QMa(1, β = 0) = KAn+1 a2 + 1, a3 + 2, . . . , an + n− 1,− ( n 2 ) − n∑ j=2 aj  . Write QMa(t, β) = ∑ k≥0 c (k) a (t)βk. The polynomials c (k) a (t) are symmetric (unimodal?) for all k. Example 5.77. Let’s take n = 5, a = (2, 1, 1, 0). One can show that the value of the Kostant partition function KA5(3, 3, 4, 4,−14) is equal to 1967. On the other hand, one has Q(2,1,1,0)(t, β)t−3 = (50, 118, 183, 233, 263, 273, 263, 233, 183, 118, 50)t + (214, 491, 738, 908, 992, 992, 908, 738, 491, 214)tβ + (365, 808, 1167, 1379, 1448, 1379, 1167, 808, 365)tβ 2 + (313, 661, 906, 1020, 1020, 906, 661, 313)tβ 3 + (139, 275, 351, 373, 351, 275, 139)tβ 4 + (29, 52, 60, 60, 52, 29)tβ 5 + (2, 3, 3, 3, 2)tβ 6, Q(2,1,1,0)(1, β) = (1967, 6686, 8886, 5800, 1903, 282, 13) = (1, 34, 279, 748, 688, 204, 13)β+1. It might be well to point out that since we know, see Theorem 5.63, that polynomials QMa(1, β) in face are polynomials of β + 1 with non-negative integer coefficients, we can treat the polynomial Q̃Ma(β) := QMa(1, β − 1) as a β-analogue of the Kostant partition function in the dominant chamber. It seems an interesting problem to find an interpretation of polynomials Q̃Ma(β) in the framework of the representation theory of Lie algebras. For example, Q̃(2,1,1,0)(β) = (1, 34, 279, 748, 688, 204, 13)β, Q̃(2,1,1,0)(β = 1) = 1967 = KA5(3, 3, 4, 4,−14). Exercises 5.78. (1) Show that Rn(t,−1) = t2(n−2)Rn−1 ( −t−1, 1 ) . (2) Show that the ratio Rn(0, β) (1 + β)n−2 is a polynomial in (β + 1) with non-negative coefficients. (3) Show that polynomial Rn(t, 1) has degree en := (n+ 1)(n− 2)/2, and Coeff[ten ]Rn(t, 1) = n−1∏ k=1 Catk. 142 A.N. Kirillov (4) Show that Q̃(n,2,3,0)(β) = ( 1, 3n+ 2, ( n+ 1 2 ) + n, ( n+ 1 3 ) + ( n 2 )) β , and KA4(n, 3, 4,−n− 7) = (n+ 2)(n+ 3)(n+ 9) 6 . Problems 5.79. (1) Assume additionally to the conditions (a′) and (b) above that x2 ij = βxij + 1 if 1 ≤ i < j ≤ n. What one can say about a reduced form of the element w0 in this case? (2) According to a result by S. Matsumoto and J. Novak [97], if π ∈ Sn is a permutation of the cyclic type λ ` n, then the total number of primitive factorizations (see definition in [97]) of π into product of n− `(λ) transpositions, denoted by Primn−`(λ)(λ), is equal to the product of Catalan numbers: Primn−`(λ)(λ) = `(λ)∏ i=1 Catλi−1. Recall that the Catalan number Catn := Cn = 1 n ( 2n n ) . Now take λ = (2, 3, . . . , n+1). Then Qn(1) = n∏ a=1 Cata = Prim(n2) (λ). Does there exist “a natural” bijection between the primitive factorizations and monomials which appear in the polynomial Qn(xij ;β)? (3) Compute in the algebra ÂCYBn(α, β) the specialization xij −→ 1, j < n, xij −→ t, 1 ≤ i < n, denoted by Pwn(t, α, β), of the reduced polynomial Psij ({xij}, α, β) corresponding to the transposition sij := ( j−2∏ k=i xk,k+1 ) xj−1,j  i∏ k=j−2 xk,k+1  ∈ ÂCYBn(α, β). For example, Ps14(t, α, β) = t5 + 3(1 + β)t4 + ((3, 5, 2)β + 3α)t3 + (2(1 + β)2 + α(5 + 4β))t2 + ((1 + β((1 + 3α) + 2α2)t+ α+ α2. On Some Quadratic Algebras 143 5.4 Reduced polynomials of certain monomials In this subsection we compute the reduced polynomials corresponding to dominant monomials of the form xm := xm1 1,2x m2 23 · · ·x mn−1 n−1,n ∈ ( ÂCYBn(β) )ab , where m = (m1 ≥ m2 ≥ · · · ≥ mn−1 ≥ 0) is a partition, and we apply the relations (a′) and (b) in the algebra (ÂCYBn(β))ab, see Definition 5.1 and Section 5.3.1, successively, starting from xm1 12 x23. Proposition 5.80. The function Zn−1 ≥0 −→ Zn−1 ≥0 , m −→ Pm(t = 1;β = 1) can be extended to a piece-wise polynomial function on the space Rn−1 ≥0 . We start with the study of powers of Coxeter elements. Namely, for powers of Coxeter elements, one has59 P(x12x23)2(β) = (6, 6, 1), P(x12x23x34)2(β) = (71, 142, 91, 20, 1) = (1, 16, 37, 16, 1)β+1, P(x12x23x34)3(β) = (1301, 3903, 4407, 2309, 555, 51, 1) = (1, 45, 315, 579, 315, 45, 1)β+1, P(x12x23x34x45)2(β) = (1266, 3798, 4289, 2248, 541, 50, 1) = (1, 44, 306, 564, 306, 44, 1)β+1, P(x12x23x34)3(β = 1) = 12527, P(x12x23x34)4(β = 0) = 26599, P(x12x23x34)4(β = 1) = 539601, P(x12x23x34x45)2(β = 1) = 12193, P(x12x23x34x45)3(β = 0) = 50000, P(x12x23x34x45)3(β = 1) = 1090199. Lemma 5.81. One has Pxn12x m 23 (β) = min(n,m)∑ k=0 ( n+m− k m )( m k ) βk = min(n,m)∑ k=0 ( n k )( m k ) (1 + β)k. Moreover, • polynomial P(x12x23···xn−1,n)m(β − 1) is a symmetric polynomial in β with non-negative coefficients. • polynomial Pxn12x m 23 (β) counts the number of (n,m)-Delannoy paths according to the number of NE steps60. Proposition 5.82. Let n and k, 0 ≤ k ≤ n, be integers. The number P(x12x23)n(x34)k(β = 0) is equal to the number of n up, n down permutations in the symmetric group S2n+k+1, see [131, A229892] and Exercises 5.30(2). Conjecture 5.83. Let n, m, k be nonnegative integers. Then the number Pxn12x m 23x k 34 (β = 0) is equal to the number of n up, m down and k up permutations in the symmetric group Sn+m+k+1. 59To simplify notation we set Pw(β) := Pw(xij = 1;β). 60Recall that a (n,m)-Delannoy path is a lattice paths from (0, 0) to (n,m) with steps E = (1, 0), N = (0, 1) and NE = (1, 1) only. For the definition and examples of the Delannoy paths and numbers, see [131, A001850, A008288] and http://mathworld.wolfram.com/DelannoyNumber.html. http://mathworld.wolfram.com/DelannoyNumber.html 144 A.N. Kirillov For example, • Take n = 2, k = 0, the six permutations in S5 with 2 up, 2 down are 12543, 13542, 14532, 23541, 24531, 34521. • Take n = 3, k = 1, the twenty permutations in S7 with 3 up, 3 down are 1237654, 1247653, 1257643, 1267543, 1347652, 1357642, 1367542, 1457632, 1467532, 1567432, 2347651, 2357641, 2367541, 2457631, 2467531, 2567431, 3457621, 3467521, 3567421, 4567321, see [131, A229892]. • Take n = 3, m = 2, k = 1, the number of 3 up, 2 down and 1 up permutations in S7 is equal to 50 = P321(0): 1237645, 1237546, . . . , 4567312. • Take n = 1, m = 3, k = 2, the number of 1 up, 3 down and 2 up permutations in S7 is equal to 55 = P132(0), as it can be easily checked. On the other hand, Px4 12x 3 23x 2 34x45 (β = 0) = 7203 < 7910, where 7910 is the number of 4 up, 3 down, 2 up and 1 down permutations in the symmetric group S11. Conjecture 5.84. Let k1, . . . , kn−1 be a sequence of non-negative integer numbers, consider monomial M := xk1 12x k2 23 · · ·x kn−1 n−1,n. Then reduced polynomial PM (β−1) is a unimodal polynomial in β with non-negative coefficients. Example 5.85. P3,2,1(β) = (1, 14, 27, 8)β+1 = P1,2,3(β), P2,3,1(β) = (1, 15, 30, 9)β+1 = P1,3,2(β), P3,1,2(β) = (1, 11, 18, 4)β+1 = P2,1,3(β), P4,3,2,1(β) = (1, 74, 837, 2630, 2708, 885, 68)β+1, P4,3,2,1(0) = 7203 = 3 · 74, P5,4,3,2,1(β) = (1, 394, 19177, 270210, 1485163, 3638790, 4198361, 2282942, 553828, 51945, 1300)β+1, P5,4,3,2,1(0) = 12502111 = 1019× 12269. Exercises 5.86. (1) Show that if n ≥ m, then xnijx m jk  xij=1=xjk = n∑ a=0 ( m+ a− 1 a )n−a∑ p=0 ( m p ) βp xm+a ik . (2) Show that if n ≥ m ≥ k, then Pxn12x m 23x k 34 (β) = Pxn12x m 23 (β) + ∑ a≥1 b,p≥0 ( m p )( k a )( a− 1 b )( n+ 1 p+ a− b )( m+ a− 1− b a ) (β + 1)p+a. In particular, if n ≥ m ≥ k, then Pxn12x m 23x k 34 (0) = ( m+ n n ) + ∑ a≥1 ( k a )( a∑ b=1 ( m+ n+ 1 m+ b )( a− 1 b− 1 )( m+ b− 1 a )) . Note that the set of relations from the item (1) allows to give an explicit formula for the polynomial PM (β) for any dominant sequence M = (m1 ≥ m2 ≥ · · · ≥ mk) ∈ (Z>0)k. Namely, PM (β + 1) = ∑ a k∏ j=2 ( mj + aj−1 − 1 aj−1 )∑ b k−1∏ j=1 ( mj+1 bj ) βbj  , On Some Quadratic Algebras 145 where the first sum runs over the following set A(M) of integer sequences a = (a1, . . . , ak−1) A(M) := {0 ≤ aj ≤ mj + aj−1, j = 1, . . . , k − 1}, a0 = 0, and the second sum runs over the set B(M) of all integer sequences b = (b1, . . . , bk−1) B(M) := ⋃ a∈A(M) {0 ≤ bj ≤ min(mj+1,mj − aj + aj−1)}, j = 1, . . . , k − 1. (3) Show that # ∣∣A(n, 1k−1 )∣∣ = n+ 1 k ( 2k + n k − 1 ) = f (n+k,k), where f (n+k,k) denotes the number of standard Young tableaux of shape (n+k, k). In particular, #\|A(1k)\| = Ck+1. (4) Let n ≥ m ≥ 1 be integers and set M = (n,m, 1k). Show that PM (xij = 1;β = 0) = n∑ p=0 m+ p+ 1 k ( m+ p− 1 p )( m+ 2k + p k − 1 ) := Pk(n,m). In particular, P1(n,m) = ( n+m n ) +m ( n+m+1 n ) , Pk(n, 1) = n+ 1 k + 1 ( 2k + 2 + n k ) , Pk(2, 2) = ( 79k2 + 341k + 360 ) (2k + 2)! k!(k + 5)! . Let us remark that Pk(n, 1) = n+ 1 n+ k + 2 ( 2(k + 1) + n k + 1 ) = F (2) k+1(n) = D(k, 1, n, 2), where the D(k, 1, n, 2) and F (2) k+1(n) are defined in Section 5.2.4. (5) Let T ∈ STY((n + k, k)) be a standard Young tableau of shape (n + k, k). Denote by r(T ) the number of integers j ∈ [1, n+ k] such that the integer j belongs to the second row of tableau T , whereas the number j + 1 belongs to the first row of T . Show that Pxn12x23···xk+1,k+2 (β − 1) = ∑ T∈STY((n+k,k)) βr(T ). (6) Let M = (m1,m2, . . . ,mk−1) ∈ Zk−1 >0 be a composition. Denote by ←− M the composition (mk−1,mk−2, . . . ,m2,m1), and set for short PM (β) := P∏k−1 i=1 x mi i,i+1 (xij = 1;β). Show that PM (β) = P←− M (β). Note that in general, Pk−1∏ i=1 x mi i,i+1 (xij ;β) 6= Pk−1∏ i=1 x mk−i i,i+1 (xij ;β). (7) Define polynomial PM (t, β) to be the following specialization xij −→ 1, i < j < n, xin −→ t, i = 1, . . . , n− 1 146 A.N. Kirillov of a polynomial Pk−1∏ i=1 x mi i,i+1 (xij ;β). Show that if n ≥ m, then Pxn12x m 23 (t, β) = m∑ j=0 ( m j )(n+m−j−1∑ k=m−1 ( k m− 1 ) tk−m+1 ) βj . See Lemma 5.31 for the case t = 1. (8) Define polynomials R̃n(t) as follows R̃n(t) := P(x12x23x34)n ( −t−1, β = −1 ) (−t)3n. Show that polynomials R̃n(t) have non-negative coefficients, and R̃n(0) = (3n)! 6(n!)3 . (9) Consider reduced polynomial Pn,2,2(β) corresponding to monomial xn12(x23x34)2 and set P̃n,2,2(β) := Pn,2,2(β − 1). Show that P̃n,2,2(β) ∈ N[β] and P̃n,2,2(1) = T (n+ 5, 3), where the numbers T (n, k) are defined in [131, A110952, A001701]. Conjecture 5.87. Let λ be a partition. The element sλ(θ (n) 1 , . . . , θ (n) m ) of the algebra 3T (0) n can be written in this algebra as a sum of(∏ x∈λ h(x) ) × dimVλ′ (gl(n−m)) × dimVλ (gl(m)) monomials with all coefficients are equal to 1. Here sλ(x1, . . . , xm) denotes the Schur function corresponding to the partition λ and the set of variables {x1, . . . , xm}; for x ∈ λ, h(x) denotes the hook length corresponding to a box x; V (gl(n)) λ denotes the highest weight λ irreducible representation of the Lie algebra gl(n). Problems 5.88. (1) Define a bijection between monomials of the form s∏ a=1 xia,ja involved in the polynomial P (xij ;β), and dissections of a convex (n+2)-gon by s diagonals, such that no two diagonals intersect their interior. (2) Describe permutations w ∈ Sn such that the Grothendieck polynomial Gw(t1, . . . , tn) is equal to the “reduced polynomial” for a some monomial in the associative quasi-classical Yang–Baxter algebra ÂCYBn(β). (3) Study “reduced polynomials” corresponding to the monomials • transposition: s1n := (x12x23 · · ·xn−2,n−1)2xn−1,n, • powers of the Coxeter element: (x12x23 · · ·xn−1,n)k, in the algebra ÂCYBn(α, β)ab. (4) Construct a bijection between the set of k-dissections of a convex (n + k + 1)-gon and “pipe dreams” corresponding to the Grothendieck polynomial G (β) π (n) k (x1, . . . , xn). As for a definition of “pipe dreams” for Grothendieck polynomials, see [78] and [42]. On Some Quadratic Algebras 147 Comments 5.89. We don’t know any “good” combinatorial interpretation of polynomials which appear in Problem 5.88(3) for general n and k. For example, Ps13(xij = 1;β) = (3, 2)β, Ps14(xij = 1;β) = (26, 42, 19, 2)β, Ps15(xij = 1;β) = (381, 988, 917, 362, 55, 2)β, Ps15(xij = 1; 1) = 2705. On the other hand, P(x12x23)2x34(x45)2(xij = 1;β) = (252, 633, 565, 212, 30, 1), that is in deciding on different reduced decompositions of the transposition s1n. one obtains in general different reduced polynomials. One can compare these formulas for polynomials Psab(xij = 1;β) with those for the β- Grothendieck polynomials corresponding to transpositions (a, b), see Comments 5.37. 5.4.1 Reduced polynomials, Motzkin and Riordan numbers In this subsection we investigate reduced polynomials associated with Coxeter element Cn = u12u23 · · ·un−1,n in commutative algebra ÂCYBn(α, β) in more detail. Recall that this algebra is generated over the ring Z[z, α, β] by the set of elements {ui,j , 1 ≤ i < j ≤ n} subject to the following relations uijujk = uikuij + ujkuik + βuik + α, i < j < k. Show that Pn(1, 1, β = −1) = Mn, where Mn denotes the n-th Motzkin number that is the number of Motzkin n-paths: paths from (0, 0) to (n, 0) in an n×n grid using only steps U = (1, 1), (1, 0) and (1,−1). It is also the number of Dyck (n+1)-paths with no steps UUU , see [131, A001006] for a wide variety of combinatorial interpretations, and vast literature concerning the Motzkin numbers. For example, P7(0, 1, β = −1) = 36 + 37 + 24 + 18 + 5 + 6 + 0 + 1 = 127 = M7. Therefore we treat the polynomials Pn(t, α, β = −1) as the (t, α)-Motzkin numbers. For example, P7(t, α, β = −1) = t7 + 6αt5 + 5αt4 + (0, 4, 14)αt 3 + (0, 3, 21)αt 2 + (0, 2, 21, 14)αt + (0, 1, 14, 21)α = t7 + α(1, 2, 3, 4, 5, 6)t + α2(14, 21, 21, 14)t + α3(21, 14)t. Therefore P7(t, 1, β = −1) = 1 + 21α+ 70α2 + 35α3, P7(1, 1, β = −1) = 127 = M7. Show that Pn(0, 1, β = −1) = A005043(n), known as the Riordan number, or Motzkin sum [131]. This number, denoted by MSn, counts the number of Motzkin paths of length n with no horizontal steps at level zero; it is also equal to the number of Dyck paths of semilenght n with no peaks at odd level, see [131, A005043] for 148 A.N. Kirillov a bit more combinatorial interpretations, and literature concerning the Motzkin sum or Riordan numbers. For example, P7(t, 1,−1) = (36, 37, 24, 18, 5, 6, 0, 1), 36 = MS7. Show that the Riordan number MSn is equal to the number of underdiagonal paths from (0, 0) to the line x = n − 2, using only steps (1, 0), (0, 1) and NE = (2, 1) and beginning with the step NE = (2, 1). Note that the number of such paths with no steps NE is equal to the Catalan number Catn−1. Let MS = {n ∈ N \|n = 22k(2r + 1) − 1, k ≥ 1, r ≥ 0} be a subset of the set of all odd integers [31]. Show that (a) MSn ≡ 1 (mod 2), if either n ≡ 0 (mod 2) or n ∈MSn, (b) MSn ≡ 0 (mod 2), if n is an odd integer and n /∈MS. Show that Pn(0, α, β) α ∣∣∣ α=0 = Nn−1(β + 1), where as before, Nn(t) denotes the Narayana polynomial. Let us set Pn(0, α, β) = ∑ k≥0 ck(β + 1)αk. Show that polynomials ck(β + 1), k ≥ 0 are symmetric (unimodal?) polynomials of the variable β + 1. Show that [131] Pn(1, 1, 0) = A052709(n+ 1). Show that [131] Pn(0, 1, 0) = A052705(n) that is the number of underdiagonal paths from (0, 0) to the line x = n − 2, using only steps R = (1, 0), V = (0, 1) and NE = (2, 1). For example, P7(0, 10) = 36 + 106 + 120 + 64 + 15 + 1 = 342 = A052705(7). Show that [131] ∂ ∂α Pn(t, α, β) ∣∣∣ α=0, β=0, t=1 = A05775(n− 1), that is the number of paths in the half-plane x ≥ 0 from (0, 0) to (n− 1, 2) or (n− 1,−3), and consisting of steps U = (1, 1), D = (1,−1) and H = (1, 0). For example, l.h.s. = 106 + 130 + 99 + 48 + 5 + 6 = 427 = A05775(6). Let us set Pn(t, α, β = 1) := ∑ k,l≥0 c (n) k,l t kαl. Show that (a) n∑ k=1 c (n) k,n−kt kαn−k = (t+ α)n−1, (b) c (n) k,n−k−1 = (k + 1) ( n− 1 k + 2 ) , 0 ≤ k ≤ n− 3, (c) c (n) 1,0 = c (n) 0,0 + (−1)n−1, n ≥ 3. On Some Quadratic Algebras 149 5.4.2 Reduced polynomials, dissections and Lagrange inversion formula Let {ai, bi, βi, αi, 1 ≤ i ≤ n − 1} be a set of parameters, consider non commutative algebra generated over the ring Z[{ai, bi, βi, αi}1≤i≤n−1] by the set of generators {uij , 1 ≤ i < j ≤ n} subject to the set of relations uijujk = aiuikuij + biujkuik + βuik + αi, 1 ≤ i < j < k ≤ n. Consider reduced expression Rn({uij}1≤i<j≤n) in the above algebra which corresponds to the “Coxeter element” Cn := u12u23 · · ·un−1,n. Note that the reduced expression Rn({uij}) is a linear combination of noncommutative mono- mials in the generators {uij , 1 ≤ i < j ≤ n} with coefficients from the ring Kn := Z[{ai, bi, βi, αi}1≤i<n]. Now to each monomial U which appears in the reduced expression Rn({uij}) we associate a dissection D := DU of a convex (n+ 1)-gon as follows. First of all let us label the vertices of a convex (n+1)-gon selected, by the numbers n+1, n, . . . , 1, written consequently and clockwise, starting from a fixed vertex, from here on named by (n+ 1)-vertex. Next, let us take a monomial U = ui1,j1 · · ·uip,jp which appears in the reduced expression Rn({uij}) with coefficient c(U) ∈ Kn. We draw diagonals in a convex (n+ 1)-gon chosen which connect vertices labeled correspondingly by numbers is and js+1, s = 1, . . . , p. It is clearly seen from the defining relations in the algebra in question when being applied to the Coxeter element above, that in fact, the diagonals we have drawn in a convex (n+1)-gon selected, do not meet at interior points of our convex (n+ 1)-gon. Therefore, to each monomial U which appears in the reduced polynomial associated with the Coxeter element Cn above, one can associate a dissecion D := DU of a convex (n + 1)-gon selected. Moreover, it is not difficult to see (e.g., cf. [58]) that there exists a natural bijection U ⇐⇒ DU between monomials which appear in the reduced expression Rn({uij}) and the set of dissections of a convex (n+ 1)-gon. As a corollary, to each dissection D := DU of a conves (n + 1)-gon one can attache the element c(D) := c(U) ∈ Kn which is equal to the coefficient in front of monomial U in the reduced expression corresponding to the Coxeter element Cn. To continue, let x = (x1, . . . , xn−1), y = (y1, . . . , yn−1) and z = (z1, . . . , zn−1) be three sets of variables, and D be a dissection of a convex (n + 1)-gon. We associate with dissection D a monomial m(D) ∈ Kn as follows m(D) := n−1∏ k=1 x n(k) k y m(k) k zr(k), where m(k) := mk(D) (resp. r(k) := rk(D) and n(k) := nk(D)) denotes the number of (convex) (mk + 2)-gons constituent a dissection D taken (resp. the number of diagonals issue out of the vertex labeled by (n+ 1); nk(D)) stands for the number of (oriented) diagonals and edges which issue out of the vertex labeled by k, k = 1, . . . , n). Therefore we associate with the reduced polynomial corresponding to the Coxeter element u12, . . . , un−1,n the following polynomial PLn(a, b,β,α,x,y, z) = ∑ D m(D)c(D), where the sum runs over all dissections D of a convex (n+ 1)-gon. 150 A.N. Kirillov To begin with we set x = 1 and consider the following specializations Bn(a,y) = PLn(a, b = 1,β = 1,α = 0,y, z = 1), Pn(z,a, b,β) = PLn(a, b,β,α = 0,y = 1, z), Show that Bn−1(a,y)) = Coefftn ( z − f(ty1, . . . , tyn) )[−1] , where f(y1, . . . , yn) = n−1∑ k=1 yku k+1, and for any formal power series g(u), d dug(u)\|u=0 = 1, we denote by g(u)[−1] the Lagrange Inverse formal power series associated with that g(u) that is a unique formal power series such that g(g[−1](u)) = u = g[−1](g(u)). Now let us recall the statement of Lagrange’s inversion theorem. Namely, let f(x) = x− ∑ k≥1 ykx k+1 be a formal power series. Then the inverse power series f [−1](u) is given by the following formula f [−1](y) = ∑ n≥1 wnu n, where wn := wn(p1, . . . , pn) = 1 n+ 1 ∑ p1,...,pn≥0∑ jpj=n ( n+ ∑ pj n, p1, . . . , pn ) yp1 1 y p2 2 · · · y pn n , where if N = m1 + · · ·+mn, then( N m1, . . . ,mn ) = N ! m1!m2! · · ·mn! denotes the multinomial coefficient. Therefore, the coefficient bn(p1, . . . , pn) := 1 n+ 1 ( n+ ∑ pj n, p1, . . . , pn ) , ∑ j jpj = n is equal to the number of dissections of a convex (n + 2)-gon which contain exactly pj convex (j + 2)-gons, see, e.g., [38]. Equivalently, the number bn(p1, . . . , pn) is equal to the number of cells of the associahedron Kn−1 which are isomorphic to the cartesian product (K0)p1 × · · · × (Kn−1)pn [90, 91]. Based on a natural and well-known bijection between the set of dissections of a convex (n + 2)-gon and the set of plane trees with (n + 1) ends and such that the all other vertices have degree at least 2, see, e.g., [134], one can readily seen that the number wn(p1, . . . , pn) defined above under constraint ∑ j jpj = n, is equal to the number of plane trees with n+ 1 ends and having pj vertices of degree j + 1. Example 5.90. For short we set Bn = PLn(a, b,β,α,x,y). (1) Quadrangular: B2 = y2 1(a1z1 + b1z1z2) + y2(β1z1 + α1). On Some Quadratic Algebras 151 (2) Pentagon: B3 = y3 1 ( a2 1z1 + a1b1z1 + a2b 2 1z1z2 + a1b1z1z3 + b21b2z1z2z3 ) + y1y2 ( 2a1β1z1 + b1β1z1 + b21β2z1z2 + b1β1z1z3 + a1α1b1α1 + α1z3 ) + y3 ( β1α1 + β2 1z1 + b21α2z1 ) . (3) Hexagon: B4 = y4 1 (( a3 1 + 2a2 1b1 + a1a2b 2 1 + a1b 2 1b2 ) z1 + a2 1b1b2z1z2 + a2b 3 1b2z1z2 + a1a3b 2 1z1z3 + a2 1b1z1z4 + a1b 2 1z1z4 + a3b 3 1b 2 2z1z2z3 + a2b 2 1b2z1z2z4 + a1b 2 1b3z1z3z4 + b31b 2 2b3z1z2z3z4 ) + y2 1y2 ( a2 1α1 + 2a1b1α1 + a2b 2 1α1 + b21b2α1 + (3a2 1bβ1 + 4a1b1β1 + a2b 2 1β1 + b21 + b2β1 + a1b 2 1β2)z1 + a2b 2 1β2z1z2 + b31b2β2z1z2 + a2b 3 1β2z1z2 + a1b 2 1b3z1z3 + a1b1β1z1z3 + a3b 2 1z1z3 + b21β1z1z4 + a1b1β1z1z4 + b21b2β3z1z2z3 + b31b2β2z1z2z4 + b21b3β1z1z3z4 ) + y1y3 ( a1β1α1 + 2b1β1α1 + ( 2a1β 2 1 + 2b1β 2 1a1b 2 1α3 + a2b 2 1α2 + b31b2α2 ) z1 + b31b2α3z1z2 + b31β 2 2z1z2 + b31α3z1z4 + b3α1z3z3z4 + a3α1z3 + a1α1z4 + b1α1z4 + β1α1z4 ) + y2 2 ( a1β1α1 + b21β2α2 + ( b21β1β2 + a1β 2 1 + a1β 2 1α2 ) z1 + β3α1z3 + b1β1β3z1z3 ) + y4(α1α3 + β2 1α1 + b21α1α2 ( b21β1α2 + b31β2α2 + β3 1 + b21β1α3 ) z1 ) . Special cases. Generalized Schröder or Lagrange polynomials: Pn(a, b,β,y, z) = Bn ∣∣ α=0 . For example, P4(a, b,y) = y4 1 (( a3 1 + 2a2 1b1 + a1a2b 2 1 + a1b 2 1b2 ) z1 + a2 1b1b2z1z2 + a2b 3 1b2z1z2 + a1a3b 2 1z1z3 + a2 1b1z1z4 + a1b 2 1z1z4 + a3b 3 1b 2 2z1z2z3 + a2b 2 1b2z1z2z4 + a1b 2 1b3z1z3z4 + b31b 2 2b3z1z2z3z4 ) + y2 1y2 (( 3a2 1bβ1 + 4a1b1β1 + a2b 2 1β1 + b21 + b2β1 + a1b 2 1β2 ) z1 + a2b 2 1β2z1z2 + b31b2β2z1z2 + a2b 3 1β2z1z2 + a1b 2 1b3z1z3 + a1b1β1z1z3 + a3b 2 1z1z3 + b21β1z1z4 + a1b1β1z1z4 + b21b2β3z1z2z3 + b31b2β2z1z2z4 + b21b3β1z1z3z4 ) + y1y3 ( 2a1β 2 1 + 2b1β 2 1 + b31β 2 2z1z2 + b1β 2 1z1z4 ) + y2 2 (( b21β1β2 + a1β 2 1 ) z1 + b1β1β3z1z3 ) + y4β 3 1z1. After the specialization ai = bi = βi = zi = 1, i = 1, 2, 3, 4, one will obtain P4(a = 1, b = 1,β = 1,y, z = 1) = 14y4 1 + 21y2 1y2 + 6y1y3 + 3y2 2 + y4. Generalized Narayana polynomials: Pn(a, b,y, z) = Bn ∣∣ α=0 β=0 , Pn(a, b,y, z) = y4 1 (( a3 1 + 2a2 1b1 + a1a2b 2 1 + a1b 2 1b2 ) z1 + a2 1b1b2z1z2 + a2b 3 1b2z1z2 + a1a3b 2 1z1z3 + a2 1b1z1z4 + a1b 2 1z1z4 + a3b 3 1b 2 2z1z2z3 + a2b 2 1b2z1z2z4 + a1b 2 1b3z1z3z4 + b31b 2 2b3z1z2z3z4 ) . Generalized Motzkin–Schröder polynomials: MSn(a, b,y, z) = Bn ∣∣ a=0 . 152 A.N. Kirillov For example, MS4(a, b,y, z) = y1 1y2 ( a2 1α1 + 2a1b1α1 + a2b 2 1α1 + b21b2α1 ) + y1y3(a1β1α1 + 2b1β1α1) + y2 2 ( a1β1α1 + b21β2α2 ) + y4 ( α1α3 + b21α1α2 + β2 1α1 ) . Generalized Motzkin polynomials: Mn(b,y, z) = Bn ∣∣ a=0 β=0 . For example, M4(b,y, z) = y4 1b 3 1b 2 2b3z1z2z3z4 + y2 1y2b 2 1b2α1 + y1y3 ( b31b2α2 + b1α1z4 + b31b2z1z3 + b31α2z1z4 + b3α1z3z4 ) + y4 ( α2α3 + b21α1α2 ) . Generalized Motzkin–Riordan polynomials: MRn(a, b,β,α,y) = Bn ∣∣ z=0 . Generalized Riordan polynomials: RIn(b,α,y) = Bn ∣∣ z=0>a=0 β=0 . For example, RI4(b,α,y) = y2 1y2b 2 1b2α1 + y4 ( α2α3 + b21α1α2 ) . Let us set Bn(y1, . . . , yn) = Bn(a = 1, b = 1,β = 1,y). Let β be a new parameter. Show that B(1, β, . . . , βn−1) = G (β) 1×w(n−1) 0 (1, . . . , 1︸︷︷︸ n ), where G (β) w (X) denotes the β-Grothendieck polynomial corresponding to a permutation w ∈ Sn. In particular, Bn(1, . . . , 1︸︷︷︸ n ) = Schn, where Schn denotes the n-th Schröder number, that is the numbers of paths from (0, 0) to (2n, 0), using only steps northeast U = (1, 1) or or D = (1,−1)) or double H = (2, 0), that never fall below the x-axis. Assume that n is devisible by an integer d ≥ 1. Show that if y = (yj = δj+1,d), then Bn(0, . . . , 0, 1︸︷︷︸ d−1 , 0, . . . , 0) = FC (d+1) n/d , where FCp m denotes the Fuss–Catalan number, see, e.g., [134], and [131, A001764] for a variety of combinatorial interpretations the Fuss–Catalan numbers FC (3) n . More generally, let 2 < d1 < · · · < dk be a sequence of integers, and set y = (δi+1,dj , 1 ≤ j ≤ k). Show that the specialization Bn(y) counts the number of dissections of a convex (n+ 2)-gon on parts which are convex (d + 2)-gons, where each d belongs to the set {d1, . . . , dk}. We would like to point out that the polynomials FS(d) n := Coeffynd ( Pnd(a, b,β,y = (δi+1,d), z) ) . can be treated as a multi-parameter analogue of the Fuss–Catalan numbers FC (d+1) n . Colored dissections [127]. A colored dissection of a convex polygon is a dissection where each (d+ 1)-gon appearing in the dissection can be colored by one of bd possible colors61, d ≥ 2 [127]. 61We assume that if bd = 0, then the dissection in question doesn’t contain parts which are (d+ 1)-gons. On Some Quadratic Algebras 153 Show [127] that if b2, . . . , bn be a sequence of non-negative integers, Bn(b2, . . . , . . . , bn) is equal to the number of colored dissections of a convex (n+ 2)-gon. Consider the specialization yi = i− 1, i = 1, . . . , n. Show that Bn(y) := SL(0, 1, . . . , n− 1) = Fine(n+ 1), where Fine(m) denotes the m-th Fine number, that is the number of ordered rooted trees with m edges having root of even degree [131, A000957]. Therefore, the Fine number Fine(n + 1) counts the number of dissections of a convex (n+ 2)-gon such that each (d+ 3)-gon appearing in the dissection can be colored by d possible colors, d ≥ 1. Consider the specialization y3k+1 = 1, y3k+2 = 0, y3k+3 = −1, k ≥ 0. Show that Bn(y1, . . . , yn) = Mn, where Mn denotes the n-th Motzkin number [131, A001006]. Recall that it is the number of ways to draw any number of nonintersecting chord joining n labeled points on a circle. The number Mn is also equals to the number of Motzkin paths, that is paths from (0, 0) to (n, n) in the n× n grid using only steps U = (1, 1), H = (1, 0) and D = (1,−1), see [131, A001006] for references and a wide variety of combinatorial interpretations of Motzkin’s numbers. Consider the specialization y3k+1 = 0, y3k+2 = (−1)k, y3k+3 = (−1)k, k ≥ 0. Show that Bn(y1, . . . , yn) = MSn, where MSn denotes the Motzkin sum or Riordan number [131, A005043]. Recall that it is the number of Motzkin paths of length n with no horizontal steps H = (1, 0) at level zero, see [131, A005043] for references and a wide variety of combinatorial interpretations of Riordan’s numbers. Consider the specialization y2k+1 = (−1)k, y2k = (−1)k+1, k ≥ 0. Show that [131] Bn(y1, . . . , yn) = A052709(n), that is the number of underdiagonal lattice paths from (0, 0) to (n−1, n−1) and such that each step is either H = (1, 0), V = (0, 1), or D = (2, 1). Consider specialization yk = (−1)k n! k! , k ≥ 1. Show that Bn(y1, . . . , yn) = nn−2, that is the number of parking functions, see, e.g., [55, 134] and the literature quoted therein. Consider the specialization yk = n! k! . Show that [131] Bn(y1, . . . , yn) = A052894(n), where A052894(n) denotes the number of Schröder trees62. A Appendixes A.1 Grothendieck polynomials Definition A.1. Let β be a parameter. The Id-Coxeter algebra IdCn(β) is an associative algebra over the ring of polynomials Z[β] generated by elements 〈e1, . . . , en−1〉 subject to the set of relations 62Schröder trees have been introduced in a paper by W.Y.C. Chen [23]. Namely, these are trees for which the set of subtrees at any vertex is endowed with the structure of ordered partition. Recall that an ordered partition of a set in which the blocks are linearly ordered [23]. 154 A.N. Kirillov • eiej = ejei if \|i− j\| ≥ 2, • eiejei = ejeiej if \|i− j\| = 1, • e2 i = βei 1 ≤ i ≤ n− 1. It is well-known that the elements {ew, w ∈ Sn} form a Z[β]-linear basis of the algebra IdCn(β). Here for a permutation w ∈ Sn we denoted by ew the product ei1ei2 · · · ei` ∈ IdCn(β), where (i1, i2, . . . , i`) is any reduced word for a permutation w, i.e., w = si1si2 · · · si` and ` = `(w) is the length of w. Let x1, x2, . . . , xn−1, xn = y, xn+1 = z, . . . be a set of mutually commuting variables. We assume that xi and ej commute for all values of i and j. Let us define hi(x) = 1 + xei, Ai(x) = i∏ a=n−1 ha(x), i = 1, . . . , n− 1. Lemma A.2. One has (1) addition formula: hi(x)hi(y) = hi(x⊕ y), where we set (x⊕ y) := x+ y + βxy; (2) Yang–Baxter relation: hi(x)hi+1(x⊕ y)hi(y) = hi+1(y)hi(x⊕ y)hi+1(x). Corollary A.3. (1) [hi+1(x)hi(x), hi+1(y)hi(y)] = 0. (2) [Ai(x), Ai(y)] = 0, i = 1, 2, . . . , n− 1. The second equality follows from the first one by induction using the addition formula, whereas the fist equality follows directly from the Yang–Baxter relation. Definition A.4 (Grothendieck expression). Gn(x1, . . . , xn−1) := A1(x1)A2(x2) · · ·An−1(xn−1). Theorem A.5 ([42]). The following identity Gn(x1, . . . , xn−1) = ∑ w∈Sn G(β) w (Xn−1)ew holds in the algebra IdCn ⊗ Z[x1, . . . , xn−1]. Definition A.6. We will call polynomial G (β) w (Xn−1) as the β-Grothendieck polynomial corre- sponding to a permutation w. Corollary A.7. (1) If β = −1, the polynomials G (−1) w (Xn−1) coincide with the Grothendieck polynomials in- troduced by Lascoux and M.-P. Schützenberger [86]. (2) The β-Grothendieck polynomial G (β) w (Xn−1) is divisible by x w(1)−1 1 . On Some Quadratic Algebras 155 (3) For any integer k ∈ [1, n − 1] the polynomial G (β−1) w (xk = q, xa = 1, ∀ a 6= k) is a poly- nomial in the variables q and β with non-negative integer coefficients. Sketch of proof. It is enough to show that the specialized Grothendieck expression Gn(xk = q, xa = 1, ∀ a 6= k) can be written in the algebra IdCn(β − 1)⊗ Z[q, β] as a linear combination of elements {ew}w∈Sn with coefficients which are polynomials in the variables q and β with non- negative coefficients. Observe that one can rewrite the relation e2 k = (β − 1)ek in the following form ek(ek + 1) = βek. Now, all possible negative contributions to the expression Gn(xk = q, xa = 1, ∀ a 6= k) can appear only from products of a form ca(q) := (1 + qek)(1 + ek) a. But using the Addition formula one can see that (1 + qek)(1 + ek) = 1 + (1 + qβ)ek. It follows by induction on a that ca(q) is a polynomial in the variables q and β with non-negative coefficients. � Definition A.8. • The double β-Grothendieck expression Gn(Xn, Yn) is defined as follows Gn(Xn, Yn) = Gn(Xn)Gn(−Yn)−1 ∈ IdCn(β)⊗ Z[Xn, Yn]. • The double β-Grothendieck polynomials {Gw(Xn, Yn)}w∈Sn are defined from the decom- position Gn(Xn, Yn) = ∑ w∈Sn Gw(Xn, Yn)ew of the double β-Grothendieck expression in the algebra IdCn(β). More details about β-Grothendieck and related polynomials can be found in [71, 84]. A.2 Cohomology of partial f lag varieties Let n = n1 + · · · + nk, ni ∈ Z≥1∀i, be a composition of n, k ≥ 2. For each j = 1, . . . , k define the numbers Nj = n1 + · · ·+ nj , N0 = 0, and Mj = nj + · · ·+ nk. Denote by X := Xn1,...,nk = {x(i) a \| i = 1, . . . , k, 1 ≤ a ≤ ni} (resp. Y , . . . ) a set of variables of the cardinality n. We set deg(x (i) a ) = a, i = 1, . . . , k. For each i = 1, . . . , k define quasihomogeneous polynomial of degree ni in variables X(i) = { x (i) a \| 1 ≤ a ≤ ni } pni ( X(i), t ) = tni + ni∑ a=1 x(i) a t ni−a, and put pn1,...,nk(X, t) = k∏ i=1 pni(X (i), t). We summarize in the theorem below some well-known results about the classical and quantum cohomology and K-theory rings of type An−1 partial flag varieties F ln1,...,nk . Let q1, . . . , qk−1, deg(qi) = ni + ni+1, i = 1, . . . , k − 1, be a set of “quantum parameters”. Theorem A.9. There are canonical isomorphisms H∗(F ln1,...,nk ,Z) ∼= Z[Xn1,...,nk ] /〈 pn1,...,nk(X, t)− tn 〉 , K•(F ln1,...,nk ,Z) ∼= Z[Y ±1] /〈 pn1,...,nk(Y , t)− (1 + t)n 〉 , 156 A.N. Kirillov H∗T (F ln1,...,nk ,Z) ∼= Z[X,Y ] /〈 k∏ i=1 ni∏ a=1 (x(i) a + t)− pn1,...,nk(Y , t) 〉 , QH∗(F ln1,...,nk) ∼= Z[Xn1,...,nk , q1, . . . , qk−1] /〈 ∆n1,...,nk(X, t)− tn 〉 (cf. [4]), QH∗T (F ln1,...,nk) ∼= Z[X,Y , q1, . . . , qk−1] /〈 ∆n1,...,nk(X, t)− pn1,...,nk(Y , t) 〉 (cf. [4]), where63 ∆n1,...,nk(X, t) = det ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ pn1(X(1), t) q1 0 · · · · · · · · · 0 −1 pn2(X(2), t) q2 0 · · · · · · 0 0 −1 pn3(X(3), t) q3 0 · · · 0 ... . . . . . . . . . . . . . . . ... 0 · · · · · · 0 −1 pnk−1 (X(k−1), t) qk−1 0 · · · · · · · · · 0 −1 pnk(X(k), t) ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ . Here for any polynomial P (x, t) = r∑ j=0 bj(x)tr−j in variables x = (x1, x2, . . .), we denote by 〈P (x, t)〉 the ideal in the ring Z[x] generated by the coefficients b0(x), . . . , br(x). A similar meaning have the symbols〈 k∏ i=1 ni∏ a=1 (x(i) a + t)− pn1,...,nk(y, t) 〉 , 〈 ∆n1,...,nk(x, t)− tn 〉 and so on. Note that dim(Fn1,...,nk) = ∑ i<j ninj and the Hilbert polynomial Hilb(Fn1,...,nk , q) of the partial flag variety Fn1,...,nk is equal to the q-multinomial coefficient [ n n1,...,nk ] q , and also is equal to the q-dimension of the weight (n1, . . . , nk) subspace of the n-th tensor power (Cn)⊗n of the fundamental representation of the Lie algebra gl(n). Comments A.10. The cohomology and (small) quantum cohomology rings H∗(Fn1,...,nk ,Z) and QH∗(Fn1,...,nk ,Z), of the partial flag variety Fn1,...,nk admit yet another representations we are going to present. To start with, let as before n = n1+· · ·+nk, ni ∈ Z≥1, ∀ i, be a composition. Consider the set of variables X̂ = Xn1,...,nk−1 := {x(i) a \| 1 ≤ i ≤ na, a = 1, . . . , k − 1}, and set as before deg x (i) a = a. Note that the number of variables X̂ is equal to n− nk. To continue, let’s define elementary quasihomogeneous polynomials of degree r er ( X̂ ) = ∑ I,A x(i1) a1 · · ·x(is) as , e0 ( X̂ ) = 1, e−r ( X̂ ) = 0, r > 0, where the sum runs over sequences of integers I = (i1, . . . , is) and A = (a1, . . . , as) such that • 1 ≤ i1 < · · · < is ≤ k − 1, • 1 ≤ aj ≤ nij , j = 1, . . . , s, and r = a1 + · · ·+ as, and complete homogeneous polynomials of degree p hp ( X̂ ) = det ∣∣ej−i+1 ( X̂ )∣∣ 1≤i,j≤p. 63We prefer to use quantum parameters {qi \| 1 ≤ i ≤ k− 1} instead of the parameters {(−1)niqi \| 1 ≤ i ≤ k− 1} have been used in [4]. On Some Quadratic Algebras 157 Finally, let’s define the ideal Jn1,...,nk in the ring of polynomials Z[Xn1,...,nk−1 ] generated by polynomials hnk+1 ( X̂ ) , . . . , hn ( X̂ ) . Note that the ideal Jn1,...,nk is generated by n− nk = #(Xn1,...,nk−1 ) elements. Proposition A.11. There exists an isomorphism of rings H∗(Fn1,...,nk ,Z) ∼= Z[Xn1,...,nk−1 ]/Jn1,...,nk . In a similar way one can describe relations in the (small) quantum cohomology ring of the partial flag variety Fn1,...,nk . To accomplish this let’s introduce quantum quasihomogeneous elementary polynomials of degree j e (q) j (Xn1,...,nr) through the decomposition ∆n1,...,nr(Xn1,...,nr) = Nr∑ j=0 e (q) j (Xn1,...,nr)t Nr−j , e (q) 0 (x) = 1, e (q) −p(x) = 0, p > 0. To exclude redundant variables { x (k) a , 1 ≤ a ≤ nk } , let us define quantum quasihomogeneous Schur polynomials s (q) α (Xn1,...,nr) corresponding to a composition α = (α1 ≤ α2 ≤ · · · ≤ αp) as follows s(q) α (Xn1,...,nr) = det ∣∣e(q) j−i+αi(Xn1,...,nr) ∣∣ 1≤i,j≤p. Proposition A.12. The (small) quantum cohomology ring QH∗(Fn1,...,nk ,Z) is isomorphic to the quotient of the ring of polynomials Z[q1, . . . , qk−1] [Xn1,...,nk−1 ] by the ideal In1,...,nk−1 gene- rated by the elements gr(Xn1,...,nk−1 ) := s (q1,...,qk−1) (1nk ,r) (Xn1,...,nk−1 )− qk−1e (q1,...,qk−2) r−nk−1 (Xn1,...,nk−2 ), where nk + 1 ≤ r ≤ n. It is easy to see that the Jacobi matrix( ∂ ∂x (i) a gr(Xn1,...,nk−1 ) ) {a=1,...,k−1, 1≤i≤na nk+1≤r≤n} corresponding to the set of polynomials gr(Xn1,...,nk−1 ), nk ≤ r ≤ n, has nonzero determi- nant, and the component of maximal degree nmax := ∑ l<j ninj in the ring QH∗(Fn1,...,nk ,Z) is a Z[q1, . . . , qk−1]-module of rank one with generator Λ = k−1∏ i=1 na∏ a=1 ( x(i) a )Mi . Therefore, one can define a scalar product (the Grothendieck residue) 〈•, •〉 : HQ∗(Fn1,...,nk ,Z)×HQ∗(Fn1,...,nk ,Z) −→ Z[q1, . . . , qk−1] setting for elements f and g of degrees a and b, 〈f, h〉 = 0, if a + b 6= nmax, and 〈f, h〉 = λ(q), if a + b = nmax and fh = λ(q)Λ. It is well known that the Grothendieck pairing 〈•, •〉 is nondegenerate (for any choice of parameters q1, . . . , qk−1). 158 A.N. Kirillov Finally we state “a mirror presentation” of the small quantum cohomology ring of partial flag varieties. To start with, let n = n1 + · · · + nk, k ∈ Zge2 be a composition of size n, and consider the set Σ(n) = { (i, j) ∈ Z× Z \| 1 ≤ i ≤ Na, Ma+1 + 1 ≤ j ≤Ma, a = 1, . . . , k − 1}, where Na = n1 + · · ·+ na, N0 = 0, Nk = n Ma = na+1 + · · ·+ nk, M0 = n,Mk = 0. With these data given, let us introduce the set of variables Zn = {zi,j \| (i, j) ∈ Σ(n)}, and define “boundary conditions” as follows • zi,Ma+1 = 0, if Na−1 + 2 ≤ i ≤ Na, a = 1, . . . , k − 1, • zNa+1,j =∞, if Ma+1 + 2 ≤ j ≤Ma, a = 1, . . . , k − 1, • zNa−1+1,Ma+1 = qa, a = 1, . . . , k, where q1, . . . , qk are “quantum parameters. Now we are ready, follow [53], to define superpotential Wq,n = ∑ (p,j)∈Σ(n) ( zi,j+1 zi,j + zi,j zi+1,j ) . Conjecture A.13 (cf. [53]). There exists an isomorphism of rings QH∗[2](F ln1,...,nk ,Z) ∼= Z [ q±1 1 , . . . , q±1 k ][ Z±1 n ] /J(Wq,n), where QH∗[2](F ln1,...,nk ,Z) denotes the subring of the ring QH∗(F ln1,...,nk ,Z) generated by the elements from H2(F ln1,...,nk ,Z). J(Wq,n) stands for the ideal generated by the partial derivatives of the superpotential Wq,n: J(Wq,n) = 〈 ∂Wq ∂zi,j 〉 , (i, j) ∈ Σ(n). Note that variables {zi,j ∈ Σ(n), i 6= Na + 1, a = 0, . . . , k − 2} are redundant, whereas the variables {za,j := z−1 Na+1,j , j = 1, . . . , na, a = 0, . . . , k − 2} satisfy the system of algebraic equations. In the case of complete flag variety F ln corresponds to partition n = (1n) and the superpo- tential Wq,1n is equal to Wq,1n = ∑ 1≤i<j≤n−1 ( zi,j+1 zi,j + zi,j zi−1,j+1 ) , where we set zi,n := qi, i = 1, . . . , n. The ideal J(Wq,1n) is generated by elements ∂Wq,1n zi,j = 1 zi,j−1 + 1 zi−1,j+1 − zi,j+1 + zi−1,j−1 z2 i,j . One can check that the ideal J(Wq,1n) can be also generated by elements of the form i∑ j=0 A (i) j (q1, . . . , qn−i+1, zn−1, . . . , zn−i+1)zj−i−1 n−i = 1, A (i) 0 = q1 · · · qn−i+1, where zi := z−1 1,i , i = 1, . . . n− 1. For example, zn1 q1 · · · qn = 1, q1q2z 2 n−1 − q2zn−2 = 1, q1q2q3z 3 n−2 − 2q1q2q3zn−1zn−2zn−3 + q2q3z 2 n−3 + q3zn−4 = 1. Therefore the number of critical points of the superpotential Wq is equal to n! = dimH∗(F ln,Z), as it should be. Note also that QH∗(F ln,Z) = QH∗[2](F ln,Z). On Some Quadratic Algebras 159 A.3 Multiparamater 3-term relations algebras A.3.1 Equivariant multiparameter 3-term relations algebras Let q = {qij}1≤i 6=j≤n, qij = qji, be a collection of mutually commuting parameters and β = {βij}1≤i 6=j≤n, βij = βji and ` = {`ij}1≤i 6=j≤n, `ij = `ji, be two sets of mutually commuting variables each. Definition A.14. Denote by 3QTn(β, `, q) an associative algebra generated over the ring Z[β, `, q] by the set of generators {x1, . . . , xn} and that {uij}1≤i 6=j≤n subject to the set of relations (1) locality conditions: [xi, xj ] = 0, [uij , ukl] = 0, [xk, uij ] = 0 if i, j, k, l are pairwise distinct, (2) generalized unitarity conditions: uij + uji = βij , (3) Hecke type conditions: uijuji = −qij if i 6= j, (4) twisted 3-term relations: uijujk = ujkuik − uikuji, ujkuij = uikujk − ujiuik if i, j, k are distinct, (5) crossing relations: xiuji = −uijxj − `ij if i 6= j. As before we define the (additive) Dunkl elements to be θi = xi + ∑ j 6=i uij , i = 1, . . . , n. (A.1) It should be pointed out that the Dunkl elements do not commute with variables {xi}, {βij} and {`ij}. It is clearly seen from the defining relations listed in Definition A.14 that for any triple of distinct indices (i, j, k) the elements {xi, xj , xk, uji, uik, ujk} satisfy the twisted dynamical Yang– Baxter relations, and thus the Dunkl elements {θi}1≤i≤n generate a commutative subalgebra in the algebra 3QTn(β, `). On the other hand, one can show that the set of defining relations involve in the definition of algebra 3QTn(β, `) implies the following set of compatibility relations among the set of generators {uij} and the set of variables {βij} and {`ij} `ijujk + uij`jk + βijujkxi + uijβjkxj = ujk`ik + `ikuij + ujkβikxi + βijxiuij , if i, j, k are distinct. These relations are satisfied, for example, if either βij = β, and `ij = h, ∀ i, j for some parameters (i.e., a central elements) β and h, or variables {βij} and {`ij} satisfy the exchange relations with generators {uij}, namely, the commutativity relations [βij , ukm] = 0, [`ij , ukm] = 0 if {i, j} ∩ {k,m} = ∅ and the exchange relations βijujk = ujkβik, `ijujk = ujk`ik if k 6= i, j. It happens that in the first case, if β = 0, then the (commutative) algebra generated by additive Dunkl’s elements and elementary symmetric polynomials {ek(Xn)}1≤k≤n (resp. multiplicative Dunkl’s elements) is isomorphic to the equivariant quantum cohomology ring (resp. to the equiv- ariant quantum K-theory ring) of the type An−1 complete flag variety. In the second case a geometric interpretation of the algebra generated by Dunkl’s elements is missing. 160 A.N. Kirillov Our main objective in this section is to to describe (part of) relations among Dunkl’s element using defining relations involve in the Definition A.14 of the algebra 3QTn(β, `, q), under the following constraints `ij = hmax(i,j), h2, . . . , hn are all central. Note, that except the case βj = β and hi = hj , ∀ i, j, our assumption violates the crossing relations between the elements βij , `ij and uj,k, but nevertheless allows to compute explicitly (part of) relations among the Dunkl’s elements. We expect that an abstract algebra generated over Q[β,h] by a set of mutually commuting elements θ1, . . . , θn and elementary symmetric polynomials {ek(Xn)}1≤k≤n subject to the set of relations descending from those for Dunkl’s elements which were mentioned above, has some interesting combinatorial/geometric interpre- tations. Below we state some results concerning relations among Dunkl elements in the algebra 3QTn(β, `, q). Theorem A.15 (cf. Theorem 3.17, Section 3). Let k ≥ 1 be an integer. There exist polynomials Rk(q,h, z1, . . . , zn) ∈ Z[β, q, {hj − hi}1≤i<j≤n][Zn], Tk(β,h, z1, . . . , zn) ∈ Z[β,h][Zn]Sn such that Rk(q,h, z1, . . . , zn) = e (q+h) k (z1, . . . , zn) + monomials of total degree ≤ k − 2 w.r.t. variables {zi}1≤i≤n, Tk(β,h, z1, . . . , zn) = ek(z1, . . . , zn) + ∑ j<k cj,kej(Xn), cj,k ∈ Z[β,h], Rk(θ1, . . . , θn) = Tk(x1, . . . , xn), where e (q+h) k (z1, . . . , zn) denotes the multiparameter quantum elementary polynomial correspond- ing to the set of parameters {(q + h)} = {qij + hj}1≤i<j≤n. It is not difficult to see that the unitarity and crossing conditions imply the following relations [xi + xj , ukl] = 0 = [xixj , ukl], [x2 i , ukl] = 0 are valid for all indices i 6= j, k 6= l. As a consequence of these relations one can deduce that the all symmetric polynomials ek(Xn) := ek(x1, . . . , xn), k = 1, . . . , n, belong to the center of the algebra 3QTn(q,h), and therefore one has [θi, ek(Xn)] = 0 for all i and k. Let us denote by QH(β,h) a commutative subalgebra in the algebra 3QTn(β,h) generated by the elementary symmetric polynomials {ek(Xn)}1≤k≤n and the Dunkl elements {θi}1≤i≤n. It is an interesting problem to give a geometric/cohomological interpretation of the commutative algebra QH(β,h). We don’t know any geometric interpretation of that commutative algebra, except the special case [75] β = 0, hj = 1, ∀ j, qij := qiδi+1,j . (A.2) Proposition A.16 ([75]). Under assumptions (A.2), the algebra QH(0,0) isomorphic to the equivariant quantum cohomology QH∗T (F ln) of the complete f lag variety F ln. Examples A.17. Let us list the relations among the Dunkl elements in the algebra 3QTn(β,h) for n = 3, 4, and βj = β, ∀j. (1) e1(θ1, . . . , θn) = e1(Xn) + ( n 2 ) β, On Some Quadratic Algebras 161 (2) e (q+h) 2 (θ1, . . . , θn) = e2(Xn) + (n− 1)βe1(Xn) + n(n− 1)(n− 2)(3n− 1) 24 β2, n ≥ 3, (3) e (q+h) 3 (θ1, θ2, θ3) = e3(X3) + h3β, e (q+h) 3 (θ1, θ2, θ3, θ4) = e3(X4) + βe2(X4) + 2β2e1(X4) + 6β3 + β(h3 + 3h4), (4) e (q+h) 4 (θ1, θ2, θ3, θ4) + β(h4 − h3)θ4 = e4(X4) + βh4e1(X4) + 5β2h4. Note that n(n−1)(n−2)(3n−1) 24 = s(n− 2, 2) = e2(1, 2, . . . , n− 1) is equal to the Stirling number of the first kind. Conjecture A.18. The polynomial Rk(q,h, Zn), see Theorem 2.29, can be written as a poly- nomial in the variables {hij := hj − hi, 1 ≤ i < j ≤ n, z1, . . . , zn, β, qij , 1 ≤ i < j ≤ n} with nonnegative coefficients. Exercises A.19 (Pieri formula in the algebra 3Tn(0, h), [75]). Assume that β = 0 and h2 = · · · = hn = h, and denote by θ (n) i , i = 1, . . . , n the Dunkl elements (A.1) in the algebra 3Tn(0, h). Show that ek ( θ (n) 1 , . . . , θ(n) m ) = ∑ r≥0 (−h)rN(m− k, 2 r)  ∑ S⊂[1,m] I={ia}, J={ja} XSui1,j1 · · ·ui\|I\|,j\|J\|  , where N(a, 2b) = (2b− 1)!! ( a+ 2b 2b ) , XS = ∏ s∈S xs, and the second summation runs over triples of sets {S, I, J} such that S ⊂ [1,m], I ⊂ [1,m]\S, \|I\|+ \|S\|+ 2r = k, \|I\| = \|J \|, 1 ≤ ia < m < ja ≤ n and j1 ≤ · · · ≤ j\|I\|. A.3.2 Algebra 3QTn(β, h), generalized unitary case Let β = (β1, . . . , βn−1), h = (h2, . . . , hn) and {qij}1≤i<j≤n be collections of mutually commuting parameters as in the previous section. As before we define the Dunkl elements θi, i = 1, . . . , n, by the formula (A.1). It is necessary to stress that the Dunkl elements {θ}1≤i≤n do not commute in the algebra 3QTn(β,h) but satisfy a noncommutative analogue of the relations displayed in Theorem A.15. Namely, one needs to replace the both elementary polynomials ek(Zn) and the quantum multiparameter elementary polynomials e (q) k (Zn) by its noncommutative versions. Recall that the noncommutative elementary polynomial ek(Zn) is equal to∑ 1≤j1<j2<···<jk≤n zj1 zj2 · · · zjk and the noncommutative quantum multiparameters elementary polynomial e (q) k (Zn) is equal to ∑ ` ∑ 1≤i1<···<j`≤n i1<j1,...,i`<j` ek−2`(ZI∪J) ∏̀ a=1 uia,ja , where I = (i1, . . . , i`), J = (j1, . . . , j`) should be distinct elements of the set {1, . . . , n}, and ZI∪J denotes set of variables za for which the subscript a is neither one of im nor one of the jm. 162 A.N. Kirillov Example A.20. e (q+h) 2 (θ1, . . . , θn) = e2(Xn) + n−1∑ j=1 βj  e1(Xn) + ∑ 1≤a<b≤n−1 abβaβb, e (q+h) 3 (θ1, θ2, θ3, θ4) + (β3 − β1)(θ3θ4 + q34 + h4 + β2(θ1 + θ2)) + (β3 − β2)((θ1 + θ2)θ4 + q14 + q24 + 2h4 + β1θ3) = e3(X4) + β3e2(X4) + (β1β3 + β2β3 + β2 3 − β1β2)e1(X4) + (3β2 3 − β1β2)(β1 + 2β2) + β1(h3 + h4) + 2β2h4, e (q+h) 4 (θ1, θ2, θ3, θ4) + (β2h4 − β1h3)θ4 + h4(β2 − β1)θ3 = e4(X4) + β2h4e1(X4) + β2h4(2β2 + 3β3). Project A.21 (noncommutative universal Schubert polynomials). Let w ∈ Sn be a permutation and Sw(Zn) be the corresponding Schubert polynomial. (1) There exists a (noncommutative) polynomial Shw({uij}1≤i<j≤n) with non-negative integer coefficients such that the following identity Sw(θ1, . . . , θn) = Shw({uij}1≤i<j≤n) holds in the algebra 3T (0) n , where {θj}1≤j≤n are the Dunkl elements in the algebra 3T (0) n . (2) There exist polynomials Rw(β, q,h, Zn) ∈ N[β, q, hj − hi1≤i<j≤n][Zn] and Tw(β,h, Zn) ∈ Z[β,h][Zn] such that the following identity Rw(β, q,h, θ1, . . . , θn) = Tw(β,h, Xn) + Shw({uij}1≤i<j≤n) holds in the algebra 3QTn(β,h). 3) Let r ∈ Z≥2 and N = n1 + · · · + nr, nj ∈ Z≥1, ∀ j, be a composition of N , and set Nj = n1 + · · ·+nj, j ≥ 1, N0 = 0. Eliminate the Dunkl elements θ (N) Nr−1+1, . . . , θ (N) N from the set of relations among the Dunkl elements θ (N) 1 , . . . , θ (N) N in the algebra 3QTn(β,h), by the use of the degree 1, . . . , nr relations among the former. As a result one obtains a set consisting of Nr−1 relations among the Nr−1 elements θ (N) j.kj := e (q) kj ( θ (N) Nj−1+1, . . . , θ (N) Nj ) , 1 ≤ kj ≤ nj , 1 ≤ j ≤ r − 1. Give a geometric interpretation of the commutative subalgebra QHn1,...,nr(β,h) ⊂ 3QTn(β,h) generated by the set of elements θ (N) j,kj , 1 ≤ kj ≤ nj, j = 1, . . . , r − 1. A.4 Koszul dual of quadratic algebras and Betti numbers Let k be a field of zero characteristic, F (n) := k〈x1, . . . , xn〉 = ⊕ j≥0 F (n) j be the free associative algebra generated by {xi, 1 ≤ i ≤ n}. Let A = F (n)/I be a quadratic algebra, i.e., the ideal of relations I is generated by the elements of degree 2, I ⊂ F (n) 2 . Let F (n)∗ = Hom(Fn, k) =⊕ j≥0 F (n)∗ j with a multiplication induced by the rule fg(ab) = f(a)g(b), f ∈ F (n)∗ i , g ∈ F (n)∗ j , a ∈ F (n) i , b ∈ F (n) j . Let I⊥2 = {f ∈ F (n)∗ 2 , f(I2) = 0}, and denote by I⊥ the two-sided ideal in F (n)∗ generated by the set I⊥2 . Definition A.22. The Koszul (or quadratic) dual A! of a quadratic algebra A is defined to be A! := F (n)∗/I⊥. The Koszul dual of a quadratic algebra A is a quadratic algebra and (A!)! = A. On Some Quadratic Algebras 163 Examples A.23. (1) Let A = F (n) be the free associative algebra, then the quadratic dual A! = k〈y1, . . . , yn〉/(yiyj , 1 ≤ i, j ≤ n). (2) If A = k[x1, . . . , xn] is the ring of polynomials, then A! = k[y1, . . . , yn]/([yi, yj ]−, 1 ≤ i, j ≤ n), where we put by definition [yi, yj ]− = yiyj + yjyi if i 6= j, and [yi, yi] = y2 i . (3) Let A = F (n)/(f1, . . . , fr), where fi = ∑ 1≤j,k≤n aijkxjxk, i = 1, . . . , r are linear independent elements of degree 2 in F (n). Then the quadratic dual of A is equal to the quotient algebra A! = k〈y1, . . . , yn〉/J , where the ideal J = 〈g1, . . . , gs〉, s = n2 − r, is generated by elements gm = ∑ 1≤j,k≤n bmjkyjyk. The coefficients bmjk, m = l, . . . , s, 1 ≤ j, k ≤ n, can be defined from the system of linear equations ∑ 1≤j,k≤n aijkbmjk = 0, i = 1, . . . , r, m = 1, . . . , s (cf. [95, Chapter 5]). Let A = ⊕ j≥0Aj be a graded finitely generated algebra over field k. Definition A.24. The Hilbert series of a graded algebra A is defined to be the generating function of dimensions of its homogeneous components: Hilb(A, t) = ∑ k≥0 dimAkt k. The Betti–Poincaré numbers BA(n,m) of a graded algebra A are defined to be BA(i, j) := dim TorAi (k, k)j . The Poincaré series of algebra A is defined to be the generating function for the Betti numbers: PA(s, t) := ∑ i≥0,j≥0 BA(i, j)sitj . Let B is a k-module and A is a B-module. The Betti number β B(A) ij of A over B is the rank of the free module B[−j] the ith module of a minimal resolution of A over B that is βBij (A) = dimk ExtBi (A, k)j . The graded Betti series of A over B is the generating function BettiB(A, x, y) := ∑ i∈N, j∈Z βBij (A)xiy−j ∈ Z[y, y−1][[x]]. Definition A.25. A quadratic algebra A is called Koszul iff the Betti numbers BA(i, j) are equal to zero unless i = j. It is well-known that Hilb(A, t)PA(−1, t) = 1, and a quadratic algebra A is Koszul, if and only if BA(i, j) = 0 for all i 6= j. In this case Hilb(A, t) Hilb(A!,−t) = 1. Example A.26. Let F (0) n be a quotient of the free associative algebra Fn over field k with the set of generators {x1, . . . , xn} by the two-sided ideal generated by the set of elements {x2 1, . . . , x 2 n}. Then the algebra F (0) f n is Koszul, and Hilb ( F (0) n , t ) = 1+t 1−(n−1)t . We refer the reader to a nice written book by A. Polishchuk and L. Positselski [116] to read more widely in the theory of quadratic algebras, see also [94]. A.5 On relations in the algebra Z0 n Let us define algebra Z0 n to be the subalgebra in 3T 0 n generated by the elements ui,n, 1 ≤ i ≤ n−1. It is clear that Z0 n is a Sn−1-module,and well-known [46] that if one sets Hilb(Z0 k , t) := Zk(t), then Hilb ( 3T (0) n , t ) = n∏ k=2 Zk(t). There exists a natural action of algebra 3T 0 n−1 on that Z0 n. To define it, it’s convenient to put xi := ui,n, 1 ≤ i ≤ n− 1. 164 A.N. Kirillov Definition A.27 (cf. [67] and Section 2.3.4). Define operators ∇i,j , 1 ≤ i < j ≤ n − 1, which act on Z0 n, by the following rules • ∇i,j(xk) = 0 if k 6= i, j, • ∇i,j(xi) = xixj ,∇i,j(xj) = −xjxi, • twisted Leibniz rule: ∇i,j(x · y) = ∇i,j(x) · y + si,j(x) · ∇i,j(y) for x, y ∈ Z0 n and all 1 ≤ i < j ≤ n − 1. Here si,j ∈ Sn−1 denotes the transposition that interchanges i and j and fixes each k 6= i, j. Proposition A.28. The operators ∇i,j, 1 ≤ i < j ≤ n − 1, satisfy all defining relations of algebra 3T 0 n−1. In particular, the operators ∇i,j , satisfy the Coxeter and Yang–Baxter relations: • Yang–Baxter relations: ∇i,j∇i,k∇j,k = ∇j,k∇i,k∇i,j , • Coxeter relations. Let ∇j = ∇j,j+1, 1 ≤ j ≤ n− 2, then ∇j∇j+1∇j = ∇j+1∇j∇j+1, [∇i,∇j ] = 0 if \|i− j\| ≥ 2. Therefore, for each w ∈ Sn−1 one can define the operator ∇w = ∇a1 · · · ∇al , where the sequence (a1, . . . , al) is a reduce decomposition of the element w. Denote by Rn the kernel of the epimorphism ι : Zn −→ Fn−1 given by ι(uk,n) = xk, where Fn−1 := Q〈x1, . . . , xn−1〉 denotes the free associative algebra generated by the elements x1, . . . , xn−1. There exists the decomposition Rn = ⊕ k≥2Rn,k, where Rn,k denotes the degree k part of Rn. We denote by rn,k the dimension of the space Rn,k/ n−1∑ j=1 (xj,nRn,k−1 +Rn,k−1xj,n), and put rn := (rn,2, rn,3, . . . ). Example A.29. r3 = (2, 1), r4 = (3, 3, 2), r5 = (4, 6, 8, 6, 3), r6 = (5, 10, 20, 30, 39, 40, 39, 30, 20, 10, 4). Remark A.30. The same formulas for the action of ∇i,j on Z0 n given in Definition A.27, define an action of operators ∇i,j on the free algebra Fn−1. In this way we obtain a representation of the algebra 3Tn−1 on that Fn−1, cf. Section 2.3.4. Let us denote by F̂n the quotient of the free associative algebra Fn = 〈x1, . . . , xn〉 by the two-sided ideal generated by the elements {x2 ixj − xjx2 i , 1 ≤ i, j ≤ n}. It is not difficult to see that the operators ∇i,j , 1 ≤ i < j ≤ n, define a representation of the algebra 3T 0 n on that F̂n. Note that F̂n w Fn−1 ⊗ Z[y1, y2, . . . , yn], where deg(y1) = 1, deg(yj) = 2, j = 2, . . . , n. Therefore, Hilb ( F̂n, t ) = 1 (1− t)(1− (n− 1)t)(1− t2)n−1 . On Some Quadratic Algebras 165 Conjecture A.31. The kernel Rn coincides with the two-sided ideal in the free algebra Fn−1 generated by elements of the form s∏ k=1 ∇ik,jk(x2 a) for some positive integers s and 1 ≤ a ≤ n− 1. In other words, the all relations in the algebra Z0 n are consequence of the following relations u∇w(x2 1) = 0 for some u,w ∈ Sn−1. Challenge A.32. (1) Compute the numbers rn,k. (2) Prove (or disprove) that there exists a positive integer kmax := k (n) max such that rn,kmax 6= 0, but rn,k = 0 for all integers k > k (n) max. (3) These examples suggest that there might be exist a certain symmetry rn,k = rn,kmax−k+2, if 3 ≤ k < kmax, between the numbers rn,k, and moreover, rn,kmax = rn,2 − 1. If so, how to explain these properties of the numbers rn,k? We expect that if n ≥ 4, then k (n) max = 2 ( n−2 [(n−2)/2] ) . Example A.33 (cyclic relations in the algebra Z0 n). The following relation n−1∏ j=1 ∇n−j,n−j+1 ( x2 1 ) = n∑ i=1 xi ( n∏ a=i+1 xa i−1∏ a=1 xa ) xi holds in the free algebra Fn. Therefore in the algebra Z0 n one has the following cyclic relation of the degree n and length n− 1: n−1∑ i=1 xi ( n−1∏ a=i+1 xa i−1∏ a=1 xa ) xi = 0. If n ≥ 5, then by applying to monomials of the form n−1∏ j=2 ∇n−j,n−j+1(x2 1) the action of either operators ∇a,n−1, 2 ≤ a ≤ n− 3, or those ∇a,b, 1 ≤ a ≤ b− 2 ≤ n− 4, new, more complicated relations in the algebra Z0 n, i.e., non-cyclic relations, can appear. These are relations of the length 2n and degree n+ 1 in the algebra Z0 n. Conjecturally all relations in the algebra Z0 n can be obtained by this method. Proposition A.34. rn,k = (k − 2)! ( n− 1 k − 1 ) , 2 ≤ k ≤ 5, rn,6 = 4! ( n− 1 5 ) + 3 ( n− 1 4 ) , rn,7 = 5! ( n− 1 6 ) + 40 ( n− 1 5 ) , rn,8 = 6! ( n− 1 7 ) + 430 ( n− 1 6 ) + 39 ( n− 1 5 ) . A.5.1 Hilbert series Hilb ( 3T 0 n, t ) and Hilb (( 3T 0 n )! , t ) : Examples64 Examples A.35. Hilb ( 3T 0 3 , t ) = [2]2[3], Hilb ( 3T 0 4 , t ) = [2]2[3]2[4]2, Hilb ( 3T 0 5 , t ) = [4]4[5]2[6]4, 64All computations in this section were performed by using the computer system Bergman, except computa- tions of Hilb(3T 0 6 , t) in degrees from twelfth till fifteenth. The last computations were made by J. Backelin, S. Lundqvist and J.-E. Roos from Stockholm University, using the computer algebra system aalg mainly devel- oped by S. Lundqvist. 166 A.N. Kirillov Hilb ( 3T 0 6 , t ) = (1, 15, 125, 765, 3831, 16605, 64432, 228855, 755777, 2347365, 6916867, 19468980, 52632322, 137268120, 346652740, 850296030, . . . ) = Hilb ( 3T 0 5 , t ) (1, 5, 20, 70, 220, 640, 1751, 4560, 11386, 27425, 64015, 145330, 321843, 696960, 1478887, 3080190, . . . ), Hilb ( 3T 0 7 , t ) = Hilb ( 3T 0 6 , t ) (1, 6, 30, 135, 560, 2190, 8181, 29472, 103032, 351192, 1170377, . . . ), Hilb ( 3T 0 8 , t ) = Hilb ( 3T 0 7 , t ) (1, 7, 42, 231, 1190, 5845, 27671, 127239, 571299, 2514463, Hilb (( 3T 0 3 )! , t ) (1− t) = (1, 2, 2, 1), Hilb (( 3T 0 4 )! , t ) (1− t)2 = (1, 4, 6, 2,−5,−4,−1), Hilb (( 3T 0 5 )! , t ) (1− t)2 = (1, 8, 26, 40, 19,−18,−22,−8,−1), Hilb (( 3T 0 6 )! , t ) (1− t)3 = (1, 12, 58, 134, 109,−112,−245,−73, 68, 50, 12, 1), Hilb (( 3T 0 7 )! , t ) (1− t)3 = (1,18, 136, 545, 1169, 1022,−624,−1838,−837, 312, 374, 123,18, 1). We expect that Hilb((3T 0 n)!, t) is a rational function with the only pole at t = 1 of order [n/2], and the polynomial Hilb((3T 0 n)!, t)(1− t)[n/2] has degree equals to [5n/2]− 4, if n ≥ 2. A.6 Summation and Duality transformation formulas [63] Summation formula. Let a1 + · · ·+ am = b. Then m∑ i=1 [ai] ∏ j 6=i [xi − xj + aj ] [xi − xj ]  [xi + y − b] [xi + y] = [b] ∏ 1≤i≤m [y + xi − ai] [y + xi] . Duality transformation, case N = 1. Let a1 + · · ·+ am = b1 + · · ·+ bn. Then m∑ i=1 [ai] ∏ j 6=i [xi − xj + aj ] [xi − xj ] ∏ 1≤k≤n [xi + yk − bk] [xi + yk] = n∑ k=1 [bk] ∏ l 6=k [yk − yl + bl] [yk − yl] ∏ 1≤i≤m [yk + xi − ai] [yk + xi] . Acknowledgments I would like to express my deepest thanks to Professor Toshiaki Maeno for many years fruitful collaboration. I’m also grateful to Professors Yu. Bazlov, I. Burban, B. Feigin, S. Fomin, A. Isaev, M. Ishikawa, M. Noumi, B. Shapiro and Dr. Evgeny Smirnov for fruitful discussions on different stages of writing [72]. My special thanks are to Professor Anders Buch for sending me the programs for computation of the β-Grothendieck and double β-Grothendieck polynomials. Many results and examples in the present paper have been checked by using these programs, and Professor Ole Warnaar (University of Queenslad) for a warm hospitality and a kind interest and fruitful discussions of some results from [72] concerning hypergeometric functions. These notes represent an update version of Section 5 of my notes [72], which have been designed as an extended version of [66], and are based on my talks given at65 65To save place I will mention only the Universities and Institutions which I visited and gave talks/lectures, starting from the year 2010. I want to thank the all Universities and Institutions which I visited, for warm hospitality and financial support. On Some Quadratic Algebras 167 • The Simons Center for Geometry and Physics, Stony Brook University, USA, January 2010; • Department of Mathematical Sciences at the Indiana University – Purdue University In- dianapolis (IUPUI), USA, Departmental Colloquium, January 2010; • The Research School of Physics and Engineering, Australian National University (ANU), Canberra, ACT 0200, Australia, April 2010; • The Institut de Mathématiques de Bourgogne, CNRS U.M.R. 5584, Université de Bour- gogne, France, October 2010; • The School of Mathematics and Statistics University of Sydney, NSW 2006, Australia, November 2010; • The Institute of Advanced Studies at NTU, Singapore, 5th Asia-Pacif ic Workshop on Quantum Information Science in conjunction with the Festschrift in honor of Vladimir Korepin, May 2011; • The Center for Quantum Geometry of Moduli Spaces, Faculty of Science, Aarhus Univer- sity, Denmark, August 2011; • The Higher School of Economy (HES), and The Moscow State University, Russia, Novem- ber 2011; • The Research Institute for Mathematical Sciences (RIMS), the Conference Combinatorial representation theory, Japan, October 2011; • The Korean Institute for Advanced Study (KIAS), Seoul, South Korea, May/June, 2012, August 2014; • The Kavli Institute for the Physics and Mathematics of the Universe (IPMU), Tokyo, August 2013, August 2015; • The University of Queensland, Brisbane, Australia, October–November 2013; • The University of Warwick, the University of Nottingham and the University of York, Clay Mathematics Institute, Oxford, United Kingdom, May/June 2015. 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Ch 4.7 Braid, affine braid and virtual braid groups 4.7.1 Yang–Baxter groups 4.7.2 Some properties of braid and Yang–Baxter groups 4.7.3 Artin and Birman–Ko–Lee monoids 5 Combinatorics of associative Yang–Baxter algebras 5.1 Combinatorics of Coxeter element 5.1.1 Multiparameter deformation of Catalan, Narayana and Schröder numbers 5.2 Grothendieck and q-Schröder polynomials 5.2.1 Schröder paths and polynomials 5.2.2 Grothendieck polynomials and k-dissections 5.2.3 Grothendieck polynomials and q-Schröder polynomials 5.2.4 Specialization of Schubert polynomials 5.2.5 Specialization of Grothendieck polynomials 5.3 The ``longest element'' and Chan–Robbins–Yuen polytope 5.3.1 The Chan–Robbins–Yuen polytope CRYn 5.3.2 The Chan–Robbins–Mészáros polytope Pn,m 5.4 Reduced polynomials of certain monomials 5.4.1 Reduced polynomials, Motzkin and Riordan numbers 5.4.2 Reduced polynomials, dissections and Lagrange inversion formula A Appendixes A.1 Grothendieck polynomials A.2 Cohomology of partial flag varieties A.3 Multiparamater 3-term relations algebras A.3.1 Equivariant multiparameter 3-term relations algebras A.3.2 Algebra 3QTn(,h), generalized unitary case A.4 Koszul dual of quadratic algebras and Betti numbers A.5 On relations in the algebra Zn0 A.5.1 Hilbert series Hilb(to.3Tn0,t)to. and Hilb(to.(to.3Tn0)to.!,t)to.: Examples A.6 Summation and Duality transformation formulas NK References

On Some Quadratic Algebras I 1/2: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials

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