On -Polynomials for Generalized Quantum Cluster Algebras and Gupta's Formula

We show the polynomial property of -polynomials for generalized quantum cluster algebras and obtain the associated separation formulas under a mild condition. Along the way, we obtain Gupta's formulas of -polynomials for generalized quantum cluster algebras. These formulas specialize to Gupta&#...

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Published in:	Symmetry, Integrability and Geometry: Methods and Applications
Date:	2024
Main Authors:	Fu, Changjian, Peng, Liangang, Ye, Huihui
Format:	Article
Language:	English
Published:	Інститут математики НАН України 2024
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/212340
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Cite this:	On -Polynomials for Generalized Quantum Cluster Algebras and Gupta's Formula. Changjian Fu, Liangang Peng and Huihui Ye. SIGMA 20 (2024), 080, 26 pages

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author	Fu, Changjian Peng, Liangang Ye, Huihui
author_facet	Fu, Changjian Peng, Liangang Ye, Huihui
citation_txt	On -Polynomials for Generalized Quantum Cluster Algebras and Gupta's Formula. Changjian Fu, Liangang Peng and Huihui Ye. SIGMA 20 (2024), 080, 26 pages
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description	We show the polynomial property of -polynomials for generalized quantum cluster algebras and obtain the associated separation formulas under a mild condition. Along the way, we obtain Gupta's formulas of -polynomials for generalized quantum cluster algebras. These formulas specialize to Gupta's formulas for quantum cluster algebras and cluster algebras, respectively. Finally, a generalization of Gupta's formula has also been discussed in the setting of generalized cluster algebras.
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 20 (2024), 080, 26 pages On F -Polynomials for Generalized Quantum Cluster Algebras and Gupta’s Formula Changjian FU, Liangang PENG and Huihui YE Department of Mathematics, Sichuan University, Chengdu 610064, P.R. China E-mail: changjianfu@scu.edu.cn, penglg@scu.edu.cn, yehuihuimath@outlook.com Received March 12, 2024, in final form August 25, 2024; Published online September 03, 2024 https://doi.org/10.3842/SIGMA.2024.080 Abstract. We show the polynomial property of F -polynomials for generalized quantum cluster algebras and obtain the associated separation formulas under a mild condition. Along the way, we obtain Gupta’s formulas of F -polynomials for generalized quantum cluster algebras. These formulas specialize to Gupta’s formulas for quantum cluster algebras and cluster algebras respectively. Finally, a generalization of Gupta’s formula has also been discussed in the setting of generalized cluster algebras. Key words: F -polynomial; separation formula; Fock–Goncharov decomposition; generalized quantum cluster algebra; generalized cluster algebra 2020 Mathematics Subject Classification: 13F60; 16S34; 05E16 1 Introduction Cluster algebras were invented by Fomin and Zelevinsky in [7] with the aim to provide a com- binatorial framework for the study of total positivity in algebraic groups and canonical bases of quantum groups. Since then, cluster algebras have been found deep connections with many other areas of mathematics and physics, such as discrete dynamical systems, non-commutative algebraic geometry, string theory and quiver representation theory etc., cf. [13] and the refer- ences therein. A cluster algebra is a commutative algebra endowed with a distinguished set of generators called cluster variables. These generators are gathered into overlapping sets of fixed finite cardinality, called clusters, which are defined recursively from an initial one via an operation called mutation. The first fundamental result in cluster algebras is the Lau- rent phenomenon [7], which states that every cluster variable can be expressed as a Laurent polynomial in the initial ones. A basic problem in the structure theory of cluster algebras is to find an explicit expression of the Laurent polynomial of a cluster variable. Based on the Laurent phenomenon, Fomin and Zelevinsky [8] further introduced F -polynomials and estab- lished the famous separation formulas, which give an expression of a cluster variable by its g-vector and F -polynomial. We remark that these separation formulas have played key roles not only in the structure theory of cluster algebras but also in the categorification of cluster algebras. Quantum cluster algebras were introduced by Berenstein and Zelevinsky [3], which are q- deformations of cluster algebras of geometric type. It appears naturally in the study of al- gebraic varieties arising from Lie theory. Quantum cluster algebras share almost the same structure theory as the one of cluster algebras. Among others, Berenstein and Zelevinsky [3] established the Laurent phenomenon for quantum cluster algebras. Tran [21] further proved the existence of F -polynomials and established the separation formulas for quantum cluster algebras. In their study of Teichmüller space of Riemann surface with orbifold points, Chekhov and Shapiro [4] discovered a new class of commutative algebras and formulated the so-called gener- mailto:changjianfu@scu.edu.cn mailto:penglg@scu.edu.cn mailto:yehuihuimath@outlook.com https://doi.org/10.3842/SIGMA.2024.080 2 C. Fu, L. Peng and H. Ye alized cluster algebras. By definition, the notion is a generalization of cluster algebras. Chekhov and Shapiro [4] found that the Laurent phenomenon is still true for generalized cluster algebras. Nakanishi [19] proved that a generalized cluster algebra has almost the same structure theory of a cluster algebra. In particular, he introduced F -polynomials and established the separation formulas for generalized cluster algebras. It is worth mentioning that the structure of general- ized cluster algebras also appear in many other branch of math, such as representation theory of quantum affine algebra [9], WKB analysis [12] and representation theory of finite-dimensional algebras [15, 16]. Inspired by quantum cluster algebras, it is natural to pursue a quantization of a generalized cluster algebra. However, it is not clear how to define a correct q-deformation of a generalized cluster algebra at this moment. Nevertheless, Nakanishi [18] considered the quantization of coefficients for generalized cluster algebras and discovered the quantum dilogarithms of higher degrees. Bai–Chen–Ding–Xu [1] introduced the notion of generalized quantum cluster algebra, which is a generalization of quantum cluster algebras and also a q-deformation of a special class of generalized cluster algebras. Recently, the Laurent phenomenon for generalized quantum cluster algebras has been established in [2]. The aim of this paper is to extend certain structure results of (quantum) cluster algebras to the setting of generalized (quantum) cluster algebras. Under a mild condition on the muta- tion data h, we prove the polynomial property of F -polynomials and establish the separation formulas for generalized quantum cluster algebras. The key ingredients of our approach are Fock– Goncharov decomposition of mutations and Gupta type formulas. As a byproduct, we obtain Gupta’s formulas of F -polynomials for generalized quantum cluster algebras. These formulas degenerate to the Gupta’s formulas of cluster algebras, which was discovered by Gupta [11] and recently reformulated and proved by Lin–Musiker–Nakanishi [17]. Since generalized quantum cluster algebras are not q-deformations of generalized cluster algebras with principal coefficients, the F -polynomials of a generalized quantum cluster algebra do not degenerate to F -polynomials of the associated generalized cluster algebra. Hence Gupta’s formulas for generalized quantum cluster algebras do not degenerate to Gupta’s formulas of generalized cluster algebras. Never- theless, we show the strategy of [17] can be extended to prove Gupta’s formulas for generalized cluster algebras. The paper is organized as follows. In Section 2, we recollect basic results in generalized cluster algebras and generalized quantum cluster algebras. Section 3.1 is devoted to the study of Fock–Goncharov decomposition of mutations for generalized quantum cluster algebras. In Section 3.2, we show the polynomial property of F -polynomials for generalized quantum cluster algebras and their associated Gupta’s formulas. In Section 4, Gupta’s formulas for generalized cluster algebras are discussed. 2 Preliminaries In this section, we recall definitions and basic results for generalized cluster algebras [4] and generalized quantum cluster algebras [1]. Throughout this section, we fix a positive integer n. Denote by Tn the n-regular tree whose edges are labeled by the numbers 1, . . . , n such that the n edges emanating from each vertex carry different labels. We write t k t′ to indicate that the vertices t, t′ of Tn are linked by an edge labeled by k. For an integer b, we use the notation [b]+ := max(b, 0). We also denote by [1, n] the set {1, . . . , n}. Denote by AT the transpose of a matrix A. A non-zero integer vector α ∈ Zn is sign-coherent if its entries are either all non-negative, or all non-positive. For an integer vector α = (a1, . . . , an) T ∈ Zn, we denote by [α]+ := ([a1]+, . . . , [an]+) T. It is clear that α = [α]+ − [−α]+. On F -Polynomials for Generalized Quantum Cluster Algebras and Gupta’s Formula 3 2.1 Generalized cluster algebra We follow [19]. Let P be a semifield, whose addition is denoted by ⊕. Denote by ZP the group ring of the multiplicative group P over Z and QP its fraction field. Let F be the field of rational function in n variables with coefficients in QP. Definition 2.1. A labeled seed with coefficients in P is a triplet (x,y, B) such that � B = (bij) n i,j=1 is a skew-symmetrizable integer matrix; � x = (x1, . . . , xn) is an n-tuple of algebraic independent elements of F over QP; � y = (y1, . . . , yn) is an n-tuple of elements in P. We say that x is a cluster and refer to xi, yi and B as the cluster variables, the coefficients and the exchange matrix, respectively. For a given labeled seed (x,y, B) and k ∈ [1, n], we set ŷk := yk ∏n j=1 x bjk j and denote ŷ = (ŷ1, . . . , ŷn). For an integer vector a = (a1, . . . , an) T ∈ Zn, we define xa := xa11 · · ·xann , ya := ya11 · · · yann , ŷa := ŷa11 · · · ŷann . In order to introduce the mutation in generalized cluster algebra, we need the notion of mutation data. Definition 2.2. A mutation data is a pair (r, z), where � r = (r1, . . . , rn) is an n-tuple of positive integers; � z = (zi,s)i=1,...,n;s=1,...,ri−1 is a family of elements in P satisfying the reciprocity condition: zi,s = zi,ri−s for 1 ≤ s ≤ ri − 1. Throughout this subsection, we fix a mutation data (r, z) and set zi,0 = zi,ri = 1. Now we introduce the (r, z)-mutation in generalized cluster algebras. Definition 2.3. For any seed (x,y,B) with coefficients in P and k ∈ [1, n], the (r, z)-mutation of (x,y, B) in direction k is a new seed µk(x,y, B) := (x′,y′, B′) with coefficients in P defined by the following rule: x′i =  xi if i ̸= k, x−1 k ( n∏ j=1 x [−εbjk]+ j )rk rk∑ s=0 zk,sŷ εs k rk⊕ s=0 zk,sy εs k if i = k, (2.1) y′i =  y−1 k if i = k, yi ( y [εbki]+ k )rk( rk⊕ s=0 zk,sy εs k )−bki if i ̸= k, (2.2) b′ij = { −bij if i = k or j = k, bij + rk([−εbik]+bkj + bik[εbkj ]+) else, (2.3) where ε ∈ {±1}. Remark 2.4. (1) The mutation formulas (2.1), (2.2) and (2.3) are independent of the choice of ε and µk is an involution. 4 C. Fu, L. Peng and H. Ye (2) If r = (1, . . . , 1), then the mutation formulas (2.1), (2.2) and (2.3) reduce to the mutation formulas of cluster algebras. (3) The mutation of ŷ is similar as (2.2) of y: ŷ′i =  ŷ−1 k if i = k, ŷi ( ŷ [εbki]+ k )rk ( rk∑ s=0 zk,sŷ εs k )−bki if i ̸= k. By assigning the labeled seed (x,y, B) to a root vertex t0 ∈ Tn, we obtain an (r, z)-seed pattern t 7→ Σt of (x,y, B) in the same way as cluster algebras. In particular, for each ver- tex t ∈ Tn, we have a labeled seed Σt = (xt,yt, Bt) and if t k t′ , then Σt′ = µk(Σt). We denote by xt = (x1;t, . . . , xn;t), yt = (y1;t, . . . , yn;t) and Bt = (bj;t) = (bij;t). Definition 2.5. The generalized cluster algebra A := A(t 7→ Σt) associated to the (r, z)-seed pattern t 7→ Σt of (x,y, B) is the ZP-subalgebra of F generated by X := ⋃ t∈Tn xt. The Laurent phenomenon still holds for generalized cluster algebras. Proposition 2.6 ([4, Theorem 2.5]). Each cluster variable xi;t could be expressed as a Laurent polynomial of x with coefficients in ZP. 2.2 Generalized cluster algebra with principal coefficients From now on, let y = (y1, . . . , yn) and z = (zi,s)i=1,2,...,n;s=1,2,...,ri−1 with zi,s = zi,ri−s be formal variables and P = Trop(y, z) the tropical semifield of y and z, which is the multiplicative abelian group freely generated by y and z with tropical sum ⊕ defined by(∏ i yaii ∏ i,s z ai,s i,s ) ⊕ (∏ i ybii ∏ i,s z bi,s i,s ) = (∏ i y min{ai,bi} i ∏ i,s z min{ai,s,bi,s} i,s ) , where ai, ai,s, bi, bi,s ∈ Z. Let (x,y, B) be a labeled seed with coefficients in P. Fix an (r, z)- seed pattern t 7→ Σt of (x,y, B) by assigning (x,y, B) to the vertex t0 ∈ Tn. The associated generalized cluster algebra is called a generalized cluster algebra with principal coefficients. In this case, we denote it by A• to indicate the principal coefficients. We assign two integer matrices Ct = (c1;t, . . . , cn;t) = (cij;t) n i,j=1 and Gt = (g1;t, . . . ,gn;t) = (gij;t) n i,j=1 to each vertex t ∈ Tn by the following recursion: � Ct0 = Gt0 = In; � if t k t′ ∈ Tn, then cij:t′ = { −cij;t if j = k, cij;t + rk(cik;t[εbkj;t]+ + [−εcik;t]+bkj;t) if j ̸= k, (2.4) gi;t′ =  gi;t if i ̸= k, −gk;t + rk ( n∑ j=1 [−εbjk;t]+gj;t − n∑ j=1 [−εcjk;t]+bj;t0 ) if i = k. (2.5) We remark that the recurrence formulas (2.4) and (2.5) are independent of the choice of the sign ε ∈ {±1}. We call t 7→ Ct and t 7→ Gt the (r, z)-C-pattern and (r, z)-G-pattern of the (r, z)- seed pattern of (x,y, B) respectively. The column vectors of Ct and Gt are called c-vectors On F -Polynomials for Generalized Quantum Cluster Algebras and Gupta’s Formula 5 and g-vectors of the (r, z)-seed pattern of (x,y, B), respectively. We remark that t 7→ Ct and t 7→ Gt only depend on B, r and t0 ∈ Tn. Denote by R = diag{r1, . . . , rn}. Note that both RB and BR are skew-symmetrizable. Hence we may assign (x,y, RB) and (x,y, BR) to the vertex t0 to obtain (ordinary) seed patterns of (x,y, RB) and (x,y, BR) respectively. It was proved by [19, Proposition 3.9] that the C- matrix Ct of the (r, z)-seed pattern of (x,y, B) coincide with the C-matrix of the seed pattern of (x,y, RB) at vertex t. Alternatively, R−1CtR is the C-matrix of the ordinary seed pattern of (x,y, BR) at vertex t. As a consequence, every c-vector of the (r, z)-seed pattern of (x,y, B) is sign-coherent. In this case, we also say Ct is column sign-coherent. On the other hand, the g-vectors of the (r, z)-seed pattern of (x,y, B) at vertex t coincide with the g-vectors of the ordinary seed pattern of (x,y, BR) at vertex t. Hence the G-matrix Gt is row sign-coherent, i.e., each row vector of Gt is sign-coherent. By the sign coherence of c-vectors provided in [10, Corollary 5.5], (2.5) can be rewritten as gi;t′ =  gi;t if i ̸= k, −gk;t + rk ( n∑ j=1 [−εk;tbjk;t]+gj;t ) if i = k, (2.6) where εk;t is the common sign of components of the c-vector ck;t. Similar to the ordinary seed pattern, we have the following tropical duality between C- matrices and G-matrices. Proposition 2.7 ([19, Proposition 3.21]). Let D0 be a diagonal matrix with positive integer diagonal entries such that D0RB is skew-symmetric. For each t ∈ Tn, we have D−1 0 R−1(Gt) TD0RCt = In. (2.7) Let D0R = diag { d−1 1 , . . . , d−1 n } . We denote by (−,−)D0R : Qn × Qn → Q the inner product defined by (u,v)D0R = uTD0Rv, where u,v ∈ Qn. With this notation, equation (2.7) is equivalent to (gi;t, djcj:t)D0R = δij , ∀i, j ∈ [1, n]. (2.8) Furthermore, by noticing that D0RBt is skew-symmetric, we have (u, Btv)D0R = −(Btu,v)D0R, ∀u,v ∈ Qn. Proposition 2.8. The following equality for the (r, z)-seed pattern t 7→ Σt holds: GtBt = Bt0Ct. (2.9) Proof. Since R−1CtR is the C-matrix of the seed pattern of (x,y, BR) and Gt is the G-matrix of the seed pattern of (x,y, BR), we have GtBtR = Bt0R ( R−1CtR ) by [8, equation (6.14)]. ■ For the generalized cluster algebra A• with principal coefficients, we have the strong Laurent phenomenon. Proposition 2.9 ([19, Proposition 3.3]). Each cluster variable xi;t belongs to Z[x±,y, z]. Definition 2.10. The F -polynomial Fi;t := Fi;t[y, z] of the cluster variable xi;t is defined as Fi;t[y, z] := xi;t\|x1=···=xn=1 ∈ Z[y, z]. Proposition 2.11 ([19, Theorem 3.23]). For each t ∈ Tn and i ∈ [1, n], the following formula holds: xi;t = xgi;tFi;t[ŷ, z]. 6 C. Fu, L. Peng and H. Ye Remark 2.12. Similar to ordinary cluster algebras, the following statements are equivalent for generalized cluster algebras (cf. [19, Proposition 3.19]): � every c-vectors ci;t is sign-coherent; � every F-polynomial Fi;t[y, z] has a constant term 1; � every F-polynomial Fi;t[y, z] has a unique monomial of maximal degree as a polynomial in y. Moreover its coefficients is 1 and it is divided by all the other occurring monomials as a polynomial in y. 2.3 Generalized quantum cluster algebras In this subsection, we introduce the definition and some properties of generalized quantum cluster algebras. We follow [1]. Let q be an indeterminate. Let m ≥ n be two positive integers, fix the mutation data (R,h), where R = diag{r1, . . . , rn} is a diagonal n × n matrix whose diagonal coefficients are positive integers and h = (h1; . . . ;hn) is defined as follows. For k ∈ [1, n], hk := { hk,0 ( q 1 2 ) , hk,1 ( q 1 2 ) , . . . , hk,rk ( q 1 2 )} , where hk,i ( q 1 2 ) ∈ Z [ q± 1 2 ] satisfying hk,i ( q 1 2 ) = hk,rk−i ( q 1 2 ) and hk,0 ( q 1 2 ) = hk,rk ( q 1 2 ) = 1. A compatible pair ( B̃,Λ ) consists of an integer m× n-matrix B̃ and a skew-symmetric inte- germ×m-matrix Λ such that B̃TΛ = [D 0], whereD = diag { d−1 1 , . . . , d−1 n } is a diagonal n× n matrix whose diagonal coefficients are positive integers. It is easy to see that the principal part B (i.e., the submatrix formed by the first n rows) of B̃ is skew-symmetrizable and D is a skew- symmetrizer of B. We define EB̃R k,ε as the m ×m-matrix which differs from the identity matrix only in its k-th column whose coefficients are given by ( EB̃R k,ε ) ik = { −1 if i = k, [−εbikrk]+ if i ̸= k. Denote by FRB̃ k,ε the n × n-matrix which differs from the identity matrix only in its k-th row whose coefficients are given by ( FRB̃ k,ε ) ki = { −1 if i = k, [εrkbki]+ if i ̸= k. Let k ∈ [1, n]. The mutation µk in direction k transforms the compatible pair ( B̃,Λ ) into µk ( B̃,Λ ) := ( B̃′,Λ′), where B̃′ = EB̃R k,ε B̃F RB̃ k,ε , Λ′ = ( EB̃R k,ε )T ΛEB̃R k,ε . It is straightforward to check that the first equality is equivalent to (2.3). Moreover, ( B̃′,Λ′) is a compatible pair and ( B̃′)TΛ′ = [D 0]. Fix a skew-symmetric bilinear form λ : Zm×Zm → Z. The quantum torus Tλ associated with λ is the Z [ q± 1 2 ] -algebra generated by the distinguished Z [ q± 1 2 ] -basis {x(α) \| α ∈ Zm} with mul- tiplication given by x(α)x(β) = q 1 2 λ(α,β)x(α+ β) for any α, β ∈ Zm. The quantum torus Tλ is an Ore domain and we denote by Fq := Fλ its fraction skew field, which will be the ambient field to define the generalized quantum cluster algebras. On F -Polynomials for Generalized Quantum Cluster Algebras and Gupta’s Formula 7 Definition 2.13. An (R,h)-quantum seed in Fq is a triple Σ = ( X, B̃,Λ ) , where ( B̃,Λ ) is a compatible pair and X = (X1, . . . , Xm) is an m-tuple of elements of Fq such that � X1, . . . , Xm generated Fq over Q; � XiXj = qλijXjXi, where Λ = (λij). The (labeled) set X is called a quantum cluster, X1, . . . , Xn are quantum cluster variables and Xn+1, . . . , Xm are coefficients. We define X(α) := q 1 2 ∑ i<j aiajλjiXa1 1 · · ·Xam m for any α = (a1, . . . , am)T ∈ Zm. In particular, Xi = X(ei), where e1, . . . , em is the standard Z-basis of Zm. It follows that X(α)X(β) = q 1 2 αTΛβX(α+ β) for any α, β ∈ Zm. The subalgebra of Fq generated by X(α), α ∈ Zm, is a free Z [ q± 1 2 ] -module with basis {X(α) \| α ∈ Zm}, and hence identifies with the quantum torus TΛ associated with the bilinear form Λ: Zm × Zm → Z, (α, β) 7→ αTΛβ, induced by Λ. For any β ∈ Zn, we also introduce the notation Ŷβ := X ( B̃β ) . It is straightforward to check that Ŷβ1Ŷβ2 = q 1 2 βT 1DBβ2Ŷβ1+β2 , where β1, β2 ∈ Zn. Definition 2.14. Let Σ = ( X, B̃,Λ ) be an (R,h)-quantum seed in Fq. For any k ∈ [1, n], the mutation µk in direction k transforms the seed Σ into a new triple µk(Σ) := ( X′, B̃′,Λ′), where (1) ( B̃′,Λ′) = µk ( B̃,Λ ) ; (2) X′ = (X ′ 1, . . . , X ′ m) is given by X ′ i := X′(ei) =  X(ei) if i ̸= k, rk∑ s=0 hk,s ( q 1 2 ) X(s[εbk]+ + (rk − s)[−εbk]+ − ei) if i = k, where bk is the k-th column vector of B̃ and ε ∈ {±1}. By [1, Propositions 3.6 and 3.7], the triple ( X′, B̃′,Λ′) is also an (R,h)-quantum seed in Fq and µk is an involution. For a given (R,h)-quantum seed ( X, B̃,Λ ) in Fq, we assign each vertex t ∈ Tn an (R,h)- quantum seed Σt in Fq which can be obtained from ( X, B̃,Λ ) by iterated mutations such that if t and t′ are linked by an edge labeled k, then Σt′ = µk(Σt). We call such an assignment t 7→ Σt an (R,h)-quantum seed pattern. It is clear that an (R,h)-quantum seed pattern is uniquely determined by the assignment of ( X, B̃,Λ ) to an arbitrary vertex t0 ∈ Tn. In this case, we refer to t0 the root vertex, Σt0 = ( X, B̃,Λ ) the initial (R,h)-quantum seed and X1, . . . , Xn the initial quantum cluster variables. In the following, when we fix an (R,h)-quantum seed pattern, we always denote by Σt = ( Xt, B̃t,Λt ) and Xt = (X1;t, . . . , Xm;t), B̃t = (bij;t), Λt = (λij;t). 8 C. Fu, L. Peng and H. Ye For the initial quantum seed Σt0 , we denote Xt0 := X = (X1, . . . , Xm), B̃t0 := B̃ = (bij), Λt0 := Λ = (λij). For each vertex t, we refer to Xt a quantum cluster, Xi;t (1 ≤ i ≤ n) quantum cluster variables and Xn+i;t (1 ≤ i ≤ m−n) coefficients. It is clear that for every t ∈ Tn, we have Xn+i;t = Xn+i for 1 ≤ i ≤ m− n. Definition 2.15. Given an (R,h)-quantum seed pattern t 7→ Σt, the (R,h)-quantum cluster algebra Aq(Σt0) with initial seed Σt0 = ( X, B̃,Λ ) is the Z [ q± 1 2 ][ X±1 n+1, . . . , X ±1 m ] subalgebra of Fq generated by all the quantum cluster variables Xq := {Xi;t \| 1 ≤ i ≤ n, t ∈ Tn}. Remark 2.16. The algebra Aq(Σt0) is also called the generalized quantum cluster algebra associ- ated with the (R,h)-quantum seed ( X, B̃,Λ ) . In particular, if R = diag{1, . . . , 1}, then Aq(Σt0) is a quantum cluster algebra in the sense of Berenstein and Zelevinsky [3]. On the other hand, it can be viewed as a q-deformation for a special generalized cluster algebra in the sense of Chekhov and Shapiro [4]. Fix an (R,h)-quantum seed pattern t 7→ Σt with initial seed Σt0 = ( X, B̃,Λ ) . Denote by r = (r1, . . . , rn). Recall that we have the (r, z)-C-pattern t 7→ Ct and (r, z)-G-pattern t 7→ Gt associated with B, r and t0 ∈ Tn as in Section 2.2. We introduce the (R,h)-G-pattern t 7→ G̃t for the (R,h)-quantum seed pattern t 7→ Σt as follows. For each vertex t ∈ Tn, we assign an m×m-integer matrix G̃t = (g̃1;t, . . . , g̃m;t) to t by the following recursion: (1) G̃t0 = Im; (2) if t k t′ ∈ Tn, then g̃i;t′ =  g̃i;t if i ̸= k, −g̃k;t + rk ( m∑ j=1 [−bjk;t]+g̃j;t − n∑ j=1 [−cjk;t]+bj;t0 ) if i = k. (2.10) We call G̃t the G-matrix of the (R,h)-quantum seed pattern t 7→ Σt and the column vector g̃i;t the g-vector of the quantum cluster variable Xi;t. Clearly, we have G̃t = [ Gt 0 ⋆ Im−n ] . It follows that cTi;t[D 0]g̃j;t = (ci;t,gj;t)D = d−1 i δij (2.11) for i, j ∈ [1, n]. Similar to [8, equation (6.14)], we have the following. Proposition 2.17. For each vertex t ∈ Tn, the following equation holds: G̃tB̃t = B̃t0Ct. (2.12) Proof. We prove the equality by induction on the distance between t0 and t. It is obvious for t = t0. Let t k t′ be an edge in Tn and suppose that (2.12) holds for the vertex t. We check it for t′. We first consider the k-th column, m∑ i=1 bik;t′ g̃i;t′ = ∑ i ̸=k (−bik;t)g̃i;t = − m∑ i=1 bik;tg̃i;t = − n∑ i=1 cik;tbi;t0 (by induction) = n∑ i=1 cik;t′bi;t0 . On F -Polynomials for Generalized Quantum Cluster Algebras and Gupta’s Formula 9 Now let j ̸= k, we have m∑ i=1 bij;t′ g̃i;t′ = ∑ i ̸=k bij;t′ g̃i;t′ + bkj;t′ g̃k;t′ = ∑ i ̸=k (bij;t + rk(bik;t[bkj;t]+ + [−bik;t]+bkj;t))g̃i;t + (−bkj;t) ( −g̃k;t + rk ( m∑ l=1 [−blk;t]+g̃l;t − n∑ l=1 [−clk;t]+bl;t0 )) = m∑ i=1 bij;tg̃i;t + rk[bkj;t]+ m∑ i=1 bik;tg̃i;t + rkbkj;t n∑ i=1 [−cik;t]+bi;t0 = n∑ i=1 cij;tbi;t0 + rk[bkj;t]+ n∑ i=1 cik;tbi;t0 + rkbkj;t n∑ i=1 [−cik;t]+bi;t0 = n∑ i=1 (cij;t + rk(cik;t[bkj;t]+ + [−cik;t]+bkj;t))bi;t0 = n∑ i=1 cij;t′bi;t0 . Thus (2.12) holds for the vertex t′. This completes the proof. ■ By (2.12), formula (2.10) is equivalent to the following: g̃i;t′ =  g̃i;t if i ̸= k, −g̃k;t + rk ( m∑ j=1 [−εbjk;t]+g̃j;t − n∑ j=1 [−εcjk;t]+bj;t0 ) if i = k, (2.13) where ε ∈ {±1}. By the sign-coherence of c-vectors, we have G̃t′ = G̃tE B̃tR k,εk;t , (2.14) where εk;t is common sign of components of the c-vector ck;t. If m = 2n and B̃ = [ B In ] , then (R,h)-quantum cluster algebra Aq is called a generalized quantum cluster algebra with principal coefficients. In this case, the g-vector g̃i;t is closely related to the g-vector gi;t. Lemma 2.18. Let m = 2n and B̃ = [ B In ] . Then for each vertex t ∈ Tn and 1 ≤ i ≤ n, we have g̃i;t = [ gi;t 0 ] . Proof. Note that bjk;t = c(j−n)k;t whenever n < j ≤ 2n. By the sign-coherence of c-vectors, we may choose a sign ε such that [−εc(n−j)k;t]+ = 0. Hence the recurrence formula (2.13) can be rewritten as g̃i;t′ =  g̃i;t if i ̸= k, −g̃k;t + rk n∑ j=1 [−εbjk;t]+g̃j;t if i = k. Now the result can be deduced by induction on the distance between t and t0. ■ 10 C. Fu, L. Peng and H. Ye 3 F -polynomials for generalized quantum cluster algebras Throughout this section, fix an (R,h)-quantum seed pattern t 7→ Σt with initial seed Σt0 =( X, B̃,Λ ) and keep the notation in Section 2.3. Recall that e1, . . . , em is the standard Z- basis of Zm. We denote by f1, . . . , fn the standard Z-basis of Zn. For an integer vector β = (b1, . . . , bm)T ∈ Zm, we denote by β = (b1, . . . , bn) T ∈ Zn the truncation of β. 3.1 Fock–Goncharov decomposition In this section, we introduce Fock–Goncharov decomposition of mutations for generalized quan- tum cluster algebras, which generalizes the corresponding construction of quantum cluster alge- bras in [6, 13]. For a, b ∈ Z and k ∈ [1, n], let ( rk∑ s=0 hk,s ( q 1 2 )( q b 2 z )s){a} :=  a∏ i=1 ( rk∑ s=0 hk,s ( q 1 2 )( q b(2i−1) 2 z )s) if a > 0, 1 if a = 0, −1∏ i=a ( rk∑ s=0 hk,s ( q 1 2 )( q b(2i+1) 2 z )s)−1 if a < 0. It is easy to verify that( rk∑ s=0 hk,s ( q 1 2 )( q b 2 z )s){a+a′} = ( rk∑ s=0 hk,s ( q 1 2 )( q b 2x )s){a} ∣∣∣∣ x=qa′bz ( rk∑ s=0 hk,s ( q 1 2 )( q b 2 z )s){a′} . (3.1) Let Tt be the quantum torus associated with Λt, i.e., the Z [ q± 1 2 ] -subalgebra of Fq generated by {Xt(α) \| α ∈ Zm}. It is an Ore domain. Denote by Ft the fraction skew field of Tt. Let t k t′ be an edge in Tn. The mutation µk in direction k yields a unique Z [ q± 1 2 ] -algebra isomorphism µk;t : Ft′ → Ft such that µk;t(Xt′(ei)) =  Xt(ei) if i ̸= k, rk∑ s=0 hk,s ( q 1 2 ) Xt(s[εbk]+ + (rk − s)[−εbk]+ − ei) if i = k. Recall that for α ∈ Zn, we have Ŷα t = Xt ( B̃tα ) . For k ∈ [1, n], we also denote Ŷk;t := Ŷfk t . Lemma 3.1. For k ∈ [1, n], t ∈ Tn and α ∈ Zn, there is a unique Z [ q± 1 2 ] -algebra homomor- phism ψk;t ( Ŷα t ) : Ft → Ft such that ψk;t ( Ŷα t ) (Xt(β)) = Xt(β) ( rk∑ s=0 hk,s ( q 1 2 )( q 1 2dk Ŷα t )s)−{dk(β̄,α)D} , ∀β ∈ Zm. Proof. It suffices to show that ψk;t ( Ŷα t ) (Xt(β1)Xt(β2)) = ψk;t ( Ŷα t ) (Xt(β1))ψk;t ( Ŷα t ) (Xt(β2)) for β1, β2 ∈ Zm. We have ψk;t ( Ŷα t ) (Xt(β1)Xt(β2)) = ψk;t ( Ŷα t )( q 1 2 βt 1Λtβ2Xt(β1 + β2) ) On F -Polynomials for Generalized Quantum Cluster Algebras and Gupta’s Formula 11 = q 1 2 βt 1Λtβ2Xt(β1 + β2) ( rk∑ s=0 hk,s ( q 1 2 )( q 1 2dk Ŷα t )s)−{dk(β̄1+β̄2,α)D} = Xt(β1)Xt(β2) ( rk∑ s=0 hk,s ( q 1 2 )( q 1 2dk Ŷα t )s)−{dk(β̄1,α)+dk(β̄2,α)D} = Xt(β1)Xt(β2) ( rk∑ s=0 hk,s ( q 1 2 ) (q 1 2dk +(β̄2,α)DŶα t ) s )−{dk(β̄1,α)D} × ( rk∑ s=0 hk,s ( q 1 2 )( q 1 2dk Ŷα t )s)−{dk(β̄2,α)D} = Xt(β1) ( rk∑ s=0 hk,s ( q 1 2 )( q 1 2dk Ŷα t )s)−{dk(β̄1,α)D} ×Xt(β2) ( rk∑ s=0 hk,s ( q 1 2 )( q 1 2dk Ŷα t )s)−{dk(β̄2,α)D} = ψk;t ( Ŷα t ) (Xt(β1))ψk;t ( Ŷα t ) (Xt(β2)), where the fourth equality follows from equation (3.1). ■ Lemma 3.2. For α ∈ Zn, ψk;t ( Ŷα t ) is an isomorphism. Moreover, its inverse is given by ψk;t ( Ŷα t )−1 : Ft → Ft, Xt(β) 7→ Xt(β) ( rk∑ s=0 hk,s ( q 1 2 )( q 1 2dk Ŷα t )s){dk(β̄,α)D} . Proof. It is straightforward to show that there is a unique Z [ q± 1 2 ] -algebra homomorphism Φ: Ft → Ft which maps Xt(β) to Xt(β) ( rk∑ s=0 hk,s ( q 1 2 )( q 1 2dk Ŷα t )s){dk(β̄,α)D} for any β ∈ Zm by (3.1). Furthermore, one can show that Φ ◦ ψk;t ( Ŷα t ) (Xt(β)) = Xt(β) = ψk;t ( Ŷα t ) ◦ Φ(Xt(β)) by noticing that (Btα, α)D = 0, which implies the result. ■ Proposition 3.3. For an edge t k t′ in Tn and ε ∈ {±1}, we have µk;t = ψk;t ( Ŷε k;t )ε ◦ ϕk;t;ε, (3.2) where ϕk;t;ε is the unique Z [ q± 1 2 ] -algebra isomorphism from Ft′ to Ft taking Xt′(α) to Xt ( EB̃tR k,ε α ) for any α ∈ Zm. Proof. By definition, we have Xt′(ek) = rk∑ s=0 hk;s ( q 1 2 ) Xt(s[εbk;t]+ + (rk − s)[−εbk;t]+ − ek) = rk∑ s=0 hk;s ( q 1 2 ) Xt(sεbk;t + rk[−εbk;t]+ − ek) 12 C. Fu, L. Peng and H. Ye = rk∑ s=0 hk;s ( q 1 2 ) Xt ( sεbk;t + EB̃tR k,ε ek ) = rk∑ s=0 hk;s ( q 1 2 ) q 1 2 Λt(sεbk;t,E B̃tR k,ε ek)Xt ( EB̃tR k,ε ek ) Xt(sεbk;t) = rk∑ s=0 hk;s ( q 1 2 ) q −sε 2dk Xt ( EB̃tR k,ε ek ) Xt(sεbk;t) = Xt ( EB̃tR k,ε ek ) rk∑ s=0 hk;s ( q 1 2 ) q −sε 2dk Xt(sεbk;t) = Xt ( EB̃tR k,ε ek )( rk∑ s=0 hk;s ( q 1 2 )( q 1 2dk Xt(εbk;t) )s)−ε{−ε} = Xt ( EB̃tR k,ε ek )( rk∑ s=0 hk;s ( q 1 2 )( q 1 2dk Ŷε k;t )s)−ε{−ε} . Since ( dkεfk, E B̃tR k,ε ei ) D = −εδik, we have Xt′(ek) = ψk;t ( Ŷε k;t )ε( Xt ( EB̃tR k,ε ek )) . For j ̸= k, we have Xt′(ej) = Xt(ej) = Xt ( EB̃tR k,ε ej ) = ψk;t ( Ŷε k;t )ε( Xt ( EB̃tR k,ε ej )) . ■ Remark 3.4. When R = In, equation (3.2) specializes to the Fock–Goncharov decomposition for quantum cluster algebras, where ϕk;t;+ and ϕk;t;− are called the tropical parts of µk;t (cf. [6, Section 3.3] and [13, Section 6.3]). By setting q 1 2 = 1, it further degenerate to the Fock– Goncharov decomposition for cluster algebras, see [17, Section 2.3]. Fix a path t0 i1 t1 i2 t2 i3 · · · ik tk in Tn and denote it by i. Let εj be the common sign of components of cij ;tj−1 and c+j := εjcij ;tj−1 for j ∈ [1, k]. Remark 3.5. The definition of εj is opposite to the one in [17], where they define εj to be the common sign of components of cij ;tj Our convention of εj is the same as the one in [13], which is more convenient to study maximal green sequences and green-to-red sequences in cluster algebras [14]. Now define µt0tk := µi1;t0 ◦ µi2;t1 ◦ · · ·µik;tk−1 : Ftk → Ft0 , ψ(i) := ψi1;t0 ( Ŷ c+1 t0 )ε1 ◦ ψi2;t0 ( Ŷ c+2 t0 )ε2 ◦ · · · ◦ ψik;t0 ( Ŷ c+k t0 )εk : Ft0 → Ft0 . For each j ∈ [1, k], we also set ϕt0tj := ϕi1;t0;ε1 ◦ ϕi2;t1;ε2 ◦ · · · ◦ ϕij ;tj−1;εj : Ftj → Ft0 . It is straightforward to check that ϕt0tj takes Xtj (α) to Xt0 ( G̃tjα ) by (2.14), where α ∈ Zm. Lemma 3.6. For each j ∈ [1, k − 1], we have ϕt0tj ◦ ψij+1;tj ( Ŷ εj+1 ij+1;tj )εj+1 = ψij+1;t0 ( Ŷ c+j+1 t0 )εj+1 ◦ ϕt0tj . On F -Polynomials for Generalized Quantum Cluster Algebras and Gupta’s Formula 13 Proof. It is equivalent to show ψij+1;t0 ( Ŷ c+j+1 t0 ) ◦ ϕt0tj = { ϕt0tj ◦ ψij+1;tj ( Ŷij+1;tj ) if εj+1 = +1, ϕt0tj ◦ ψij+1;tj ( Ŷ−1 ij+1;tj ) if εj+1 = −1. For any α = (α1, . . . , αm)T ∈ Zm, ϕt0tj ◦ ψij+1;tj ( Ŷ εj+1 ij+1;tj ) (Xtj (α)) = ϕt0tj Xtj (α) rij+1∑ s=0 hij+1,s ( q 1 2 )( q 1 2dij+1 Ŷ εj+1 ij+1;tj )s−{dij+1 (ᾱ,εj+1fij+1 )D}  = Xt0 ( G̃tjα )rij+1∑ s=0 hij+1,s ( q 1 2 )( q 1 2dij+1 Xt0 ( εj+1G̃tj B̃tjfij+1 ))s−{dij+1 (ᾱ,εj+1fij+1 )D} = Xt0 ( G̃tjα )rij+1∑ s=0 hij+1,s ( q 1 2 )( q 1 2dij+1 Xt0 ( εj+1B̃t0Ctjfij+1 ))s−{dij+1 (ᾱ,εj+1fij+1 )D} = Xt0 ( G̃tjα )rij+1∑ s=0 hij+1,s ( q 1 2 )( q 1 2dij+1 Xt0 ( B̃t0c + j+1 ))s−{dij+1 (ᾱ,εj+1fij+1 )D} = Xt0 ( G̃tjα )rij+1∑ s=0 hij+1,s ( q 1 2 )( q 1 2dij+1 Ŷ c+j+1 t0 )s−{αij+1 εj+1} , where the third equality follows from (2.12). On the other hand, ψij+1;t0 ( Ŷ c+j+1 t0 ) ◦ ϕt0tj (Xtj (α)) = ψij+1;t0 ( Ŷ c+j+1 t0 )( Xt0 ( G̃tjα )) = Xt0 ( G̃tjα )rij+1∑ s=0 hij+1,s ( q 1 2 )( q 1 2dij+1 Ŷ c+j+1 t0 )s−{dij+1 (G̃tjα,c + j+1)D} = Xt0 ( G̃tjα )rij+1∑ s=0 hij+1,s ( q 1 2 )( q 1 2dij+1 Ŷ c+j+1 t0 )s−{αij+1 εj+1} (by (2.11)). This completes the proof. ■ Proposition 3.7. Keep the notation as above, we have µt0tk = ψ(i) ◦ ϕt0tk : Ftk → Ft0. Proof. By using Proposition 3.3 and Lemma 3.6, we have µt0tk = µi1;t0 ◦ µi2;t1 ◦ · · · ◦ µik;tk−1 = ψi1;t0 ( Ŷε1 i1;t0 )ε1 ◦ ϕi1;t0;ε1 ◦ ψi2;t1 ( Ŷε2 i2;t1 )ε2 ◦ ϕi2;t1;ε2 ◦ · · · ◦ ψik;tk−1 ( Ŷεk ik;tk−1 )εk ◦ ϕik;tk−1;εk = ψi1;t0 ( Ŷ c+1 t0 )ε1 ◦ ψi2;t1 ( Ŷ c+2 t0 )ε2 ◦ ϕi1;t0;ε1 ◦ ϕi2;t1;ε2 ◦ · · · ◦ ψik;tk−1 (Ŷεk ik;tk−1 )εk ◦ ϕik;tk−1;εk 14 C. Fu, L. Peng and H. Ye ... = ψi1;t0 ( Ŷ c+1 t0 )ε1 ◦ ψi2;t1 ( Ŷ c+2 t0 )ε2 ◦ · · · ◦ ψik;t0 ( Ŷ c+k t0 )εk ◦ ϕt0tk = ψ(i) ◦ ϕt0tk . ■ 3.2 F -polynomials In this section, we define F -polynomials for the generalized quantum cluster algebra Aq(Σt0) and prove their polynomial property under a mild condition. Recall that for i, j ∈ [1, n], we have Ŷi,t0Ŷj,t0 = qd −1 i bijŶj,t0Ŷi,t0 . (3.3) The quasi-commutative relations (3.3) depend only on the entries of D and the principal part B of B̃t0 . Let TDB be the quantum torus associated toDB whose underlying space is the free Z [ q 1 2 ] - module with basis {Z(α)\|α ∈ Zn}, and its multiplication is given by Z(α)Z(β) = q 1 2 αTDBβZ(α+ β). Denote by FDB the fraction skew field of TDB and Zi = Z(fi). Recall that t0 i1 t1 i2 t2 i3 · · · ik tk is a path in Tn, and εj is the common sign of components of cij ;tj−1 . For simplicity of notation, we also denote by d(j) = dij , r(j) = rij , cj = cij ;tj−1 , c+j = εjcj , ĉ+j = Bc+j , g̃j = g̃ij ;tj , gj = g̃j . We first define a set of elements {Li,j \| i, j ∈ [1, k]} of Fq by the initial condition L1,i := Ŷ c+i t0 (r(1)∑ s=0 hi1,s ( q 1 2 )( q 1 2d(1) Ŷ c+1 t0 )s)−ε1{(d(1)c+1 ,ĉ+i )D} for i ∈ [1, k] with recurrence relations Lj+1,i = Lj,i (r(j+1)∑ s=0 hij+1,s ( q 1 2 )( q 1 2d(j+1)Lj,j+1 )s)−εj+1{(d(j+1)c + j+1,ĉ + i )D} for j ∈ [1, k − 1]. Then set L1 = r(1)∑ s=0 hi1,s ( q 1 2 )( q 1 2d(1) Ŷ c+1 t0 )s , Ll+1 = r(l+1)∑ s=0 hil+1,s ( q 1 2 )( q 1 2d(l+1)Ll,l+1 )s , l ∈ [1, k − 1]. Lemma 3.8. Keep the notation as above. We have Xt0(−g̃k)µ t0 tk (Xik;tk) = −→∏ j∈[1,k] L −εj{d(j)(c+j ,gk)D} j . (3.4) On F -Polynomials for Generalized Quantum Cluster Algebras and Gupta’s Formula 15 Proof. Let ij be the sub-sequence t0 i1 t1 i2 t2 i3 · · · ij tj for j ∈ [1, k]. By Propo- sition 3.7, Xt0(−g̃k)µ t0 tk (Xik;tk) = Xt0(−g̃k)ψ(i) ◦ ϕt0tk(Xik;tk) = Xt0(−g̃k)ψ(i)(Xt0(g̃k)) = Xt0(−g̃k)ψ(i1)(Xt0(g̃k))ψ(i1)(Xt0(−g̃k))ψ(i2)(Xt0(g̃k)) · · ·ψ(ik−1)(Xt0(−g̃k))ψ(ik)(Xt0(g̃k)). We first claim that Lj,i = ψ(ij) ( Ŷ c+i t0 ) for i, j ∈ [1, k]. We prove it by induction on j. For j = 1, we have ψ(i1) ( Ŷ c+i t0 ) = ψi1:t0 ( Ŷ c+1 t0 )ε1(Ŷc+i t0 ) = Ŷ c+i t0 (r(1)∑ s=0 hi1,s ( q 1 2 )( q 1 2d(1) Ŷ c+1 t0 )s)−ε1{(ĉ+i ,d(1)c + 1 )D} = L1,i. Suppose that Lj,i = ψ(ij) ( Ŷ c+i t0 ) for any i ∈ [1, k], then ψ(ij+1) ( Ŷ c+i t0 ) = ψ(ij) ( ψij+1;t0 ( Ŷ c+j+1 t0 )εj+1 ( Ŷ c+i t0 )) = ψ(ij) (Ŷc+i t0 )(r(j+1)∑ s=0 hij+1,s ( q 1 2 )( q 1 2d(j+1) Ŷ c+j+1 t0 )s)−εj+1{(ĉ+i ,d(j+1)c + j+1)D}  = ψ(ij) ( Ŷ c+i t0 ) × (r(j+1)∑ s=0 hij+1,s ( q 1 2 )( q 1 2d(j+1) ψ(ij) ( Ŷ c+j+1 t0 ))s)−εj+1{(ĉ+i ,d(j+1)c + j+1)D} = Lj,i (r(j+1)∑ s=0 hij+1,s ( q 1 2 )( q 1 2d(j+1)Lj,j+1 )s)−εj+1{(ĉ+i ,d(j+1)c + j+1)D} = Lj+1,i. This completes the proof of the claim. A direct computation shows that Xt0(−g̃k)ψ(i1)(Xt0(g̃k)) = Xt0(−g̃k)ψi1;t0 ( Ŷ c+1 t0 )ε1(Xt0(g̃k)) = (r(1)∑ s=0 hi1,s ( q 1 2 )( q 1 2d(1) Ŷ c+1 t0 )s)−ε1{(gk,d(1)c + 1 )D} = L −ε1{(gk,d(1)c + 1 )D} 1 . For j ∈ [1, k − 1], we have ψ(ij)(Xt0(−g̃k))ψ(ij+1)(Xt0(g̃k)) = ψ(ij) ( Xt0(−g̃k)ψij+1;tj ( Ŷ c+j+1 t0 )εj+1(Xt0(g̃k)) ) = ψ(ij) (r(j+1)∑ s=0 hij+1,s ( q 1 2 )( q 1 2d(j+1) Ŷ c+j+1 t0 )s)−εj+1{(gk,d(j+1)c + j+1)D}  16 C. Fu, L. Peng and H. Ye = (r(j+1)∑ s=0 hij+1,s ( q 1 2 )( q 1 2d(j+1) ψ(ij) ( Ŷ c+j+1 t0 ))s)−εj+1{(gk,d(j+1)c + j+1)D} = (r(j+1)∑ s=0 hij+1,s ( q 1 2 )( q 1 2d(j+1)Lj,j+1 )s)−εj+1{(gk,d(j+1)c + j+1)D} = L −εj+1{(gk,d(j+1)c + j+1)D} j+1 . It follows that Xt0(−g̃k)µ t0 tk (Xik;tk) = Xt0(−g̃k)ψ(i1)(Xt0(g̃k))ψ(i1)(Xt0(−g̃k))ψ(i2)(Xt0(g̃k)) · · ·ψ(ik−1)(Xt0(−g̃k))ψ(ik)(Xt0(g̃k)) = L −ε1{(gk,d(1)c + 1 )D} 1 L −ε2{(gk,d(2)c + 2 )D} 2 · · ·L−εk{(gk,d(k)c + k )D} k . ■ Definition 3.9. There is a unique element Fik;tk := Fik;tk [Z1, . . . ,Zn] in the skew field FDB of TDB such that −→∏ j∈[1,k] L −εj{d(j)(c+j ,gk)D} j = Fik;tk [ Ŷ1;t0 , . . . , Ŷn;t0 ] . (3.5) The element Fik,tk is called the F -polynomial of Xtk(eik) whenever Fik;tk is a polynomial in Z1, . . . ,Zn. With the help of Fik;tk , we can rewrite (3.4) as Xik;tk = Xt0(g̃k)Fik;tk [ Ŷ1;t0 , . . . , Ŷn;t0 ] . Remark 3.10. Recall that the (r, z)-C-pattern and (r, z)-G-pattern are uniquely determined by B, R and t0 ∈ Tn. It follows that the element Fik;tk only depends on B, R, D and t0. The following is the main result of this section. Theorem 3.11. (1) The element Fik;tk is a Laurent polynomial in Z1, . . . ,Zn. (2) Suppose that hi,s(1) > 0 for each i ∈ [1, n] and s ∈ [1, ri − 1], then Fik;tk is a polynomial in Z1, . . . ,Zn. Proof. Since Fik;tk only depends on B, D and R, but not on coefficients, it suffices to prove the statement for a particular choice of coefficients. Let B̃• = [ B In ] be the 2n× n matrix, where In is the identity matrix. There is a (2n × 2n)-skew symmetric matrix Λ• such that ( B̃•,Λ•) is a compatible pair with ( B̃•)TΛ• = [D 0] (cf. [3, Example 0.5]). It is clear that the Z [ q± 1 2 ] - algebra generated by { Ŷα t0 := Xt0 ( B̃•α ) \| α ∈ Zn } is isomorphic to TDB. According to Lemma 3.8, there are two polynomials A(Z1, . . . ,Zn), P (Z1, . . . ,Zn) ∈ TDB with coefficients in N [ q± 1 2 ] for hi,s ( q 1 2 ) 1 and Z1, . . . ,Zn such that Fik;tk ( Ŷ1;t0 , . . . , Ŷn;t0 ) = A ( Ŷ1;t0 , . . . , Ŷn;t0 ) P ( Ŷ1;t0 , . . . , Ŷn;t0 )−1 . (3.6) By [2, Theorem 3.1], Fik;tk ( Ŷ1;t0 , . . . , Ŷn;t0 ) is also a Laurent polynomial in X1;t0 , . . . ,X2n;t0 . Hence all of Fik;tk ( Ŷ1;t0 , . . . , Ŷn;t0 ) , A ( Ŷ1;t0 , . . . , Ŷn;t0 ) and P ( Ŷ1;t0 , . . . , Ŷn;t0 ) are Laurent polynomials in X1;t0 , . . . ,X2n;t0 . Taking the Newton polytopes of both sides of (3.6) as the Laurent polynomials in X1;t0 , . . . ,X2n;t0 , we obtain New(Fik;tk)+New(P ) = New(A), where the 1Here we temporary regard hi,s ( q 1 2 ) as a variable. On F -Polynomials for Generalized Quantum Cluster Algebras and Gupta’s Formula 17 sum is the Minkowski sum. By the definition of Minkowski sum, we conclude that every exponent vector of Fik;tk is a Q-linear combination of exponents vectors of A and P . In particular, the exponent vectors of Fik;tk ( Ŷ1;t0 , . . . , Ŷn;t0 ) as a Laurent polynomial in X1;t0 ,. . . , X2n;t0 can be expressed as Q-linear combinations of B̃•f1, . . . , B̃ •fn. However, the last n coordinates of the vectors B̃•fi is given by the standard basis f1, . . . , fn ∈ Zn, so each exponent vector must be a Z-linear combination of B̃•f1, . . . , B̃ •fn. This prove that Fik;tk ( Ŷ1;t0 , . . . , Ŷn;t0 ) is a Laurent polynomial in Ŷ1;t0 , . . . , Ŷn;t0 . Hence, Fik;tk [Z1, . . . ,Zn] is a Laurent polynomial in Z1, . . . ,Zn. Since A(Z1, . . . ,Zn), P (Z1, . . . ,Zn) have coefficients in N [ q± 1 2 ] for hi,s ( q 1 2 ) and Z1, . . . ,Zn. By the assumption hi,s(1) > 0 for i ∈ [1, n] and s ∈ [1, ri − 1], by setting q 1 2 = 1 does not shrink New ( Fik;tk ( Ŷ1;t0 , . . . , Ŷn;t0 )) . That is New ( Fik;tk ( Ŷ1;t0 , . . . , Ŷn;t0 )) = New ( Fik;tk ( Ŷ1;t0 , . . . , Ŷn;t0 ) \| q 1 2=1 ) . By Proposition 2.11 and Lemma 2.18, we conclude that Fik;tk ( Ŷ1;t0 , . . . , Ŷn;t0 ) \| q 1 2=1 = Fik;tk(ŷ, z)\|zi,s=hi,s(1),i∈[1,n],s∈[1,ri−1], where Fik;tk(y, z) is the F -polynomial of the cluster variable xik;tk of the corresponding general- ized cluster algebra A• with principal coefficients. Thus New ( Fik;tk ( Ŷ1;t0 , . . . , Ŷn;t0 )) does not contain any points with negative coordinates. It follows that Fik;tk ( Ŷ1;t0 , . . . , Ŷn;t0 ) is a poly- nomial in Ŷ1;t0 , . . . , Ŷn;t0 . Hence, Fik;tk [Z1, . . . ,Zn] is a polynomial in Z1, . . . ,Zn. ■ As direct consequences of Theorem 3.11 and Lemma 3.8, we have the following. Corollary 3.12 (separation formula). Suppose that hj,s(1) > 0 for each j ∈ [1, n] and s ∈ [1, rj − 1]. For each i ∈ [1, n] and t ∈ Tn, let Fi;t[Z1, . . . , Zn] be the associated F -polynomial of Xi;t and g̃i;t the g-vector of Xi;t. We have Xi;t = Xt0(g̃i;t)Fi;t ( Ŷ1;t0 , . . . , Ŷn;t0 ) . Corollary 3.13. Suppose that hj,s(1) > 0 for each j ∈ [1, n] and s ∈ [1, rj−1]. For each i ∈ [1, n] and t ∈ Tn, let Fi;t[Z1, . . . , Zn] be the associated F -polynomial of Xi;t. We have (1) There is a unique monomial q 1 2 aZ(f), a ∈ Z, in Fi;t[Z1, . . . , Zn] such that it is divisible by all the other monomials in Fi;t[Z1, . . . , Zn]; (2) Fi;t[Z1, . . . , Zn] has constant term 1. Remark 3.14. If R = In, the assumption in Theorem 3.11 is automatically satisfied. Therefore, Theorem 3.11, Corollaries 3.12 and 3.13 reduce to the corresponding results for quantum cluster algebras. Remark 3.15. In the setting of cluster algebras and quantum cluster algebras of skew-symmet- ric type, F -polynomials have the positivity property, i.e., each F -polynomial has non-negative coefficients (cf. [5, 10]). Thus if one expects that F -polynomials for generalized quantum cluster algebras still have the positivity property, then we have to assume that hi,s ( q 1 2 ) has non-negative coefficients for each i and s, which will imply that hi,s(1) > 0. Remark 3.16. We call the equation (3.5) Gupta’s formula for F -polynomials of generalized quantum cluster algebras. When R = In, it specializes to Gupta’s formula for quantum cluster algebras. It further specializes to Gupta’s formula [11, 17] for cluster algebras by setting q 1 2 = 1. In particular, let R = In and set q 1 2 = 1, we have L1,i = ŷc+i ( 1 + ŷc+1 )−(d(1)c1,ĉ + i )D for i ∈ [1, k], 18 C. Fu, L. Peng and H. Ye Lj+1,i = Lj,i(1 + Lj,j+1) −(d(j+1)cj+1,ĉ + i )D for j ∈ [1, k − 1], L1 = 1 + ŷc+1 , Ll+1 = 1 + Ll,l+1 for l ∈ [1, k − 1]. By direct computation, Ll+1 = 1 + Ll−1,l+1(1 + Ll−1,l) −(d(l)cl,ĉ + l+1)D = 1 + Ll−2,l+1(1 + Ll−2,l) −(d(l−1)cl−1,ĉ + l+1)D(1 + Ll−1,l) −(d(l)cl,ĉ + l+1)D ... = 1 + L1,l+1 l∏ p=2 (1 + Lp−1,p) −(d(p)cp,ĉ + l+1)D = 1 + ŷc+l+1 ( 1 + ŷc+1 )−(d(1)c1,ĉ + l+1)D l∏ p=2 (1 + Lp−1,p) −(d(p)cp,ĉ + l+1)D = 1 + ŷc+l+1 l∏ p=1 (Lp) −(d(p)cp,ĉ + l+1)D . Hence we obtain Gupta’s formula for cluster algebras (cf. [17]): Fik;tk = ∏k j=1 L −(d(j)cj ,gk)D j , where Fik;tk is the ik-th F -polynomial at vertex tk. We remark the above formula is slightly different from the one in [17] due to the convention on εj . The following example suggests that Theorem 3.11 (2) may hold without the assumption that hi,s(1) > 0. Example 3.17 (type G2). Let A(3, 1) be the generalized quantum cluster algebras associated to the initial (R,h)-quantum seed (X, B,Λ), where B = [ 0 1 −1 0 ] and Λ = [ 0 1 −1 0 ] , R = diag{3, 1} and h = ( 1, h ( q 1 2 ) , h ( q 1 2 ) , 1; 1, 1 ) , h ( q 1 2 ) ∈ Z [ q± 1 2 ] . By assigning (X, B,Λ) to the vertex t0 ∈ T2, we obtain an (R,h)-quantum seed pattern t 7→ Σt. Fix a path t0 1 t1 2 t2 1 t3 2 t4 1 t5 2 t6 1 t7 2 t8 . By calcula- tions, we have g1;t1 = [ −1 3 ] , g2;t2 = [ −1 2 ] , g1;t3 = [ −2 3 ] , g2;t4 = [ −1 1 ] , g1;t5 = [ −1 0 ] , g2;t6 = [ 0 −1 ] , g1;t7 = [ 1 0 ] , g2;t8 = [ 0 1 ] . Moreover, Xt1(e1) = Xt0(g1;t1) ( q− 3 2 Ŷ3e1 t0 + h ( q 1 2 ) q−1Ŷ2e1 t0 + h ( q 1 2 ) q− 1 2 Ŷe1 t0 + 1 ) , Xt2(e2) = Xt0(g2;t2) ( q− 1 2 Ŷ3e1+e2 t0 + q− 3 2 Ŷ3e1 t0 + h ( q 1 2 ) q−1Ŷ2e1 t0 + h ( q 1 2 ) q− 1 2 Ŷe1 t0 + 1 ) , Xt3(e1) = Xt0(g1;t3) ( q−6Ŷ6e1 t0 + h ( q 1 2 )( q− 11 2 + q− 9 2 ) Ŷ5e1 t0 + [ h ( q 1 2 )( q−5 + q−3 ) + h ( q 1 2 )2 q−4 ] Ŷ4e1 t0 + [ q− 9 2 + q− 3 2 + h ( q 1 2 )( q− 7 2 + q− 5 2 ) Ŷ3e1 t0 ] + [h ( q 1 2 )( q−3 + q−1 ) + h ( q 1 2 )2 q−2 ] Ŷ2e1 t0 + h ( q 1 2 )( q− 3 2 + q− 1 2 ) Ŷe1 t0 + ( q− 11 2 + q− 9 2 + q− 7 2 ) Ŷ6e1+e2 t0 + h ( q 1 2 )( q− 9 2 + 2q− 7 2 + q− 5 2 ) Ŷ5e1+e2 t0 + h ( q 1 2 )( q− 7 2 + q− 5 2 + q− 3 2 + h ( q 1 2 ) q− 5 2 ) Ŷ4e1+e2 t0 + ( q− 5 2 + q− 3 2 + q− 1 2 + h ( q 1 2 )2 q− 3 2 ) Ŷ3e1+e2 t0 + h ( q 1 2 ) q− 1 2 Ŷ2e1+e2 t0 On F -Polynomials for Generalized Quantum Cluster Algebras and Gupta’s Formula 19 + ( q−4 + q−3 + q−2 ) Ŷ6e1+2e2 t0 + h ( q 1 2 )( q− 5 2 + q− 3 2 ) Ŷ5e1+2e2 t0 + h ( q 1 2 ) q−1Ŷ4e1+2e2 t0 + q− 3 2 Ŷ6e1+3e2 t0 + 1 ) , Xt4(e2) = Xt0(g2;t4) ( q− 3 2 Ŷ3e1 t0 + h ( q 1 2 ) q−1Ŷ2e1 t0 + h ( q 1 2 ) q− 1 2 Ŷe1 t0 + ( q− 3 2 + q− 1 2 ) Ŷ3e1+e2 t0 + h ( q 1 2 ) q− 1 2 Ŷ2e1+e2 t0 + q− 1 2 Ŷ3e1+2e2 t0 + 1 ) , Xt5(e1) = Xt0(g1;t5) ( q− 3 2 Ŷ3e1 t0 + h ( q 1 2 ) q−1Ŷ2e1 t0 + h ( q 1 2 ) q− 1 2 Ŷe1 t0 + ( q− 5 2 + q− 3 2 + q− 1 2 ) Ŷ3e1+e2 t0 + h ( q 1 2 )( q− 3 2 + q− 1 2 ) Ŷ2e1+e2 t0 + h ( q 1 2 ) q− 1 2 Ŷe1+e2 t0 + ( q− 5 2 + q− 3 2 + q− 1 2 ) Ŷ3e1+2e2 t0 + h ( q 1 2 ) q−1Ŷ2e1+2e2 t0 + q− 3 2 Ŷ3e1+3e2 t0 + 1 ) , Xt6(e2) = Xt0(g2;t6) ( q− 1 2 Ŷe2 t0 + 1 ) , Xt7(e1) = Xt0(g1;t7) = Xt0(e1), Xt8(e2) = Xt0(g2;t8) = Xt0(e2), where e1, e2 is the standard Z-basis of Z2. In particular, each Fij ;tj is a polynomial in Ŷ1, Ŷ2 for any choice of h ( q 1 2 ) . Moreover, it has non-negative coefficients when regarding h ( q 1 2 ) as a variable. This phenomenon will be studied in a forthcoming paper. 4 Gupta’s formula for generalized cluster algebras As we have seen in the proof of Theorem 3.11 that F -polynomials of a generalized quantum cluster algebra do not specialize to F -polynomials of a generalized cluster algebra with principal coefficients, hence Gupta’s formula (3.5) does not reduce to a formula of F -polynomials for a generalized cluster algebra. However, we can prove the Gupta’s formula for F -polynomials of generalized cluster algebras following Sections 4.2 and 4.3 (cf. [17]). Keep notation as in Section 2.2. Recall that (x,y, B) is a labeled seed in F with coefficients in P = Trop(y, z). By assigning (x,y, B) to the vertex t0 ∈ Tn, we obtain an (r, z)-seed pattern t 7→ Σt. 4.1 Fock–Goncharov decomposition In this section, we establish Fock–Goncharov decomposition for generalized cluster algebras with principal coefficients, which generalizes the corresponding construction for cluster alge- bras [6, 20]. Let t k t′ in Tn and k ∈ [1, n], denote by εk;t the common sign of components of the c-vector ck;t. As before, we also set c+k;t := εk;tck;t and ĉk;t = Bck;t. By the sign-coherence of ck;t, the exchange formula (2.1) at xk;t can be simplified as xk;t′ = x−1 k;t ( n∏ j=1 x [−εk;tbjk;t]+ j;t )rk rk∑ s=0 zk,sŷ εk;ts k;t . Let QP(xt) be the field of rational functions in xt with coefficients in QP. The mutation µk;t yields an isomorphism µk;t : QP(xt′) → QP(xt) which maps xi;t′ to xi;t for i ̸= k and maps xk;t′ to x−1 k;t ( n∏ j=1 x [−εk;tbjk;t]+ j;t )rk ( rk∑ s=0 zk,sŷ εk;ts k;t ) . We introduce two homomorphisms of fields as follows: τk;t : QP(x′ t) −→ QP(xt), xi;t′ 7→  xi;t if i ̸= k, x−1 k;t ( n∏ j=1 x [−εk;tbjk;trk]+ j;t ) if i = k, 20 C. Fu, L. Peng and H. Ye and ρk;t : QP(xt) −→ QP(xt), xi;t 7→ xi;t ( rk∑ s=0 zk,sŷ εk;ts k;t )−δi,k . It is clear that µk;t = ρk;t ◦ τk;t and µk;t ◦ µk;t′ = 1. Moreover, we have the following by direct computation. Lemma 4.1. The following relation holds: τk;t ◦ τk;t′ = 1. We further introduce an automorphism qk;t of QP(x) as follows qk;t : QP(x) −→ QP(x), xm 7−→ xm ( rk∑ s=0 zk,s ( ŷc+k;t )s)−(m,dkck;t)D0R , ∀m ∈ Zn. By computation, we have qk;t(ŷ n) = ŷn ( rk∑ s=0 zk,s ( ŷc+k;t )s)(n,dkĉk;t)D0R , ∀n ∈ Zn. (4.1) Lemma 4.2. The following formulas hold: qk;t ( xgi;t ) = xgi;t ( rk∑ s=0 zk,s ( ŷc+k;t )s)−δi;k , (4.2) qk;t′ ◦ qk;t = 1. (4.3) Proof. Formula (4.2) is a direct consequence of (2.8). For any m ∈ Zn, qk;t′ ◦ qk;t(xm) = qk;t′ xm ( rk∑ s=0 zk,s ( ŷc+k;t )s)−(m,dkck;t)D0R  = xm ( rk∑ s=0 zk,s ( ŷ c+ k;t′ )s)−(m,dkck;t′ )D0R ×  rk∑ s=0 zk,s ŷc+k;t ( rk∑ s=0 zk,s ( ŷ c+ k;t′ )s)(c+k;t,dkĉk;t′ )D0R s−(m,dkck;t)D0R . Note that D0RB is skew-symmetric, ( c+k;t, dkĉk;t′ ) D0R = −dkεk;t(ck;t)TD0RBck;t = 0. On the other hand, ck;t′ = −ck;t. Putting all of these together, we obtain that qk;t′ ◦ qk;t ( xm ) = xm. This completes the proof of (4.3). ■ From now on, we fix a path t0 i1 t1 i2 t2 i3 · · · ik tk in Tn. We define µt0tk := µi1;t0 ◦ µi2;t1 ◦ · · · ◦ µik;tk−1 : QP(xtk) → QP(xt0), τ t0tk := τi1;t0 ◦ τi2;t1 ◦ · · · ◦ τik;tk−1 : QP(xtk) → QP(xt0), qt0tk := qi1;t0 ◦ qi2;t1 ◦ · · · ◦ qik;tk−1 : QP(xt0) → QP(xt0). Lemma 4.3. For any i ∈ [1, n], the following formulas hold: µt0tk(xi;tk) = xgi;tkFi;tk(ŷ, z), (4.4) τ t0tk (xi;tk) = xgi;tk , (4.5) τ t0tk (ŷi;tk) = ŷci;tk . (4.6) On F -Polynomials for Generalized Quantum Cluster Algebras and Gupta’s Formula 21 Proof. Formula (4.4) is equivalent to the separation formula of generalized cluster algebras [19]. Formula (4.5) follows form (2.6) by induction on k, and (4.6) follows from (4.5) and (2.9). ■ Lemma 4.4. For any j ∈ [1, k] and l ∈ [1, n], we have τ t0tj ◦ ρl;tj = ql;tj ◦ τ t0 tj . (4.7) Proof. For any i ∈ [1, n], τ t0tj ◦ ρl;tj (xi;tj ) = τ t0tj xi;tj ( rl∑ s=0 zl,s ( ŷ εl;tj l;tj )s)−δi,l  = xgi;tj ( rl∑ s=0 zl,s ( ŷ εl;tj cl;tj )s)−δi,l = xgi;tj ( rl∑ s=0 zl,s ( ŷ c+l;tj )s)−δi,l = ql;tj ◦ τ t0 tj (xi;tj ), where the last equality follows from (4.2) and (4.5). ■ By applying (4.7), we have Proposition 4.5. The following decomposition holds: µt0tk = qt0tk ◦ τ t0 tk : QP(xtk) −→ QP(xt0). The following is a direct consequence of Proposition 4.5 and Lemma 4.3. Proposition 4.6. For any i ∈ [1, n], the following formula holds: qt0tk ( xgi;tk ) = xgi;tkFi;tk(ŷ, z). 4.2 Gupta’s formula Throughout this section, we fix a path t0 i1 t1 i2 t2 i3 · · · ik tk in Tn. For the simplicity of notation, for any j ∈ [1, k], we denote by d(j) := dij , r(j) := rij , cj := cij ;tj−1 , c+j := εij ;tj−1cj , ĉ + j :=Bc+j , gj := gij ;tj , where εij ;tj−1 is the common sign of components of cij ;tj−1 . We also introduce certain elements L1, . . . , Lk of F along the path, where L1 = r(1)∑ s=1 zi1,s ( ŷc+1 )s , Ll = r(l)∑ s=0 zil,s ŷc+l l−1∏ j=1 L −(ĉ+l ,d(j)cj)D0R j s for 2 ≤ l ≤ k. The following is the Gupta’s formula of F -polynomials for generalized cluster algebras. Theorem 4.7. The following formula holds: Fik;tk(ŷ, z) = k∏ j=1 L −(gk,d(j)cj)D0R j . (4.8) Before giving the proof of Theorem 4.7, we prepare the following lemma. Lemma 4.8. For 1 ≤ i ≤ m ≤ k, the following formula holds: qt0ti ( ŷc+m ) = ŷc+m ( i∏ j=1 L −(ĉ+m,d(j)cj)D0R j ) . (4.9) 22 C. Fu, L. Peng and H. Ye Proof. We prove this formula by induction on i. For i = 1, by (4.1), we get qt0t1 ( ŷc+m ) = ŷc+m (r(1)∑ s=0 zi1,s ( ŷc+1 )s)(c+m,d(1)ĉi1;t0 )D0R = ŷc+m(L1) −(ĉ+m,d(1)c1)D0R . Now suppose that qt0tl−1 ( ŷc+m ) = ŷc+m l−1∏ j=1 L −(ĉ+m,d(j)cj)D0R j  . Then, qt0tl ( ŷc+m ) = qt0tl−1 ( qil;tl−1 ( ŷc+m )) = qt0tl−1 ( ŷc+m ( r(l)∑ s=0 zil,s(ŷ c+l )s )−(ĉ+m,d(l)cl)D0R ) = ŷc+m l−1∏ j=1 L −(ĉ+m,d(j)cj)D0R j  ×  r(l)∑ s=0 zil,sŷ sc+l l−1∏ j=1 L −s(ĉ+l ,d(j)cj)D0R j −(ĉ+m,d(l)cl)D0R = ŷc+m l−1∏ j=1 L −(ĉ+m,d(j)cj)D0R j  (Ll) −(ĉ+m,d(l)cl)D0R = ŷc+m  l∏ j=1 L −(ĉ+m,d(j)cj)D0R j  . ■ Proof of Theorem 4.7. For 2 ≤ l ≤ k, we have qt0tl (x gk) qt0tl−1 (xgk) = qt0tl−1 ( qil;tl−1 (xgk) xgk ) = qt0tl−1 (( r(l)∑ s=0 zil,s ( ŷc+l )s)−(gk,d(l)cl)D0R ) = ( r(l)∑ s=0 zil,s ( qt0tl−1 ( ŷc+l ))s)−(gk,d(l)cl)D0R =  r(l)∑ s=0 zil,s ŷc+l l−1∏ j=1 L −(ĉ+l ,d(j)cj)D0R j s−(gk,d(l)cl)D0R (by (4.9)) = (Ll) −(gk,d(l)cl)D0R . On the other hand, qi1;t0(x gk) xgk = (r(1)∑ s=0 zi1,s ( ŷc+1 )s)−(gk,d(1)c1)D0R = L −(gk,d(1)c1)D0R 1 . On F -Polynomials for Generalized Quantum Cluster Algebras and Gupta’s Formula 23 According to Proposition 4.6, we have Fik;tk(ŷ, z) = qt0tk(x gk) xgk = qt0tk(x gk) qt0tk−1 (xgk) ◦ qt0tk−1 (xgk) qt0tk−2 (xgk) ◦ · · · ◦ qt0t1 (x gk) xgk = (Lk) −(gk,d(k)ck)D0R(Lk−1) −(gk,d(k−1)ck−1)D0R · · · (L1) −(gk,d(1)c1)D0R = k∏ j=1 L −(gk,d(j)cj)D0R j . This completes the proof. ■ Example 4.9. Let us explain Theorem 4.7 by the following simplest non-trivial example. Let r = (2, 1), z = (1, z, 1; 1, 1) and B = [ 0 1 −1 0 ] . We consider the following path in T2: t0 1 t1 2 t2 1 t3 2 t4 . By assigning the labeled seed Σ = (x,y, B) to the vertex t0, we obtain an (r, z)-seed pattern t 7→ Σt. By definition, ŷ1 = y1x −1 2 , ŷ2 = y2x1, RBt0 = [ 0 2 −1 0 ] , d(1) = d(2) = d(3) = d(4) = 1 2 , and the c-,ĉ- and g-vectors involved are as following: c1 = [ 1 0 ] , c2 = [ 2 1 ] , c3 = [ 1 1 ] , c4 = [ 0 1 ] , ĉ1 = [ 0 −1 ] , ĉ2 = [ 1 −2 ] , ĉ3 = [ 1 −1 ] , ĉ4 = [ 1 0 ] , g1 = [ −1 2 ] , g2 = [ −1 1 ] , g3 = [ −1 0 ] , g4 = [ 0 −1 ] . The relevant inner products2 are given by( ĉ+2 , d(1)c1 ) = 1, ( ĉ+3 , d(1)c1 ) = 1, ( ĉ+3 , d(2)c2 ) = 1,( ĉ+4 , d(1)c1 ) = 1, ( ĉ+4 , d(2)c2 ) = 2, ( ĉ+4 , d(3)c3 ) = 1, (g2, d(1)c1) = −1, (g3, d(1)c1) = −1, (g3, d(2)c2) = −2, (g4, d(1)c1) = 0, (g4, d(2)c2) = −1, (g4, d(3)c3) = −1. Therefore, L1 = 1 + zŷ1 + ŷ21, L2 = 1 + ŷ21 ŷ2L −1 1 , L3 = 1 + z ( ŷ1ŷ2L −1 1 L−1 2 ) + ( ŷ1ŷ2L −1 1 L−1 2 )2 , L4 = 1 + ŷ2L −1 1 L−2 2 L−1 3 . Applying Theorem 4.7, we get F1;t1 = L1 = 1 + zŷ1 + ŷ21, F2;t2 = L1L2 = 1 + zŷ1 + ŷ21 + ŷ21 ŷ2, F1;t3 = L1L 2 2L3 = 1 + zŷ1 + ŷ21 + zŷ1ŷ2 + 2ŷ21 ŷ2 + ŷ21 ŷ 2 2, F2;t4 = L2L3L4 = 1 + ŷ2. 4.3 Expansion of Gupta’s formula Following [17], we expand Gupta’s formula (4.8) into sum in this section. Let h ∈ Z and n0, . . . , nl ∈ N, we denote{ h n0, n1, . . . , nl } := ( h n0 )( n0 n1, . . . , nl ) , 2Here we omit the subscript D0R. 24 C. Fu, L. Peng and H. Ye where ( n0 n1,...,nl ) is the multinomial coefficient. In particular, if n0 ̸= n1 + · · · + nl, then{ h n0,n1,...,nl } = 0. It easy to see that if h > 0, then{ h n0, n1, . . . , nl } = ( h h− n0, n1, . . . , nl ) . Lemma 4.10. Let 1 + ∑l i=1 aiz i be a polynomial in z with coefficients in a field F and h ∈ Z, we have the following expansion formula:( 1 + l∑ i=1 aiz i )h = ∑{ h n0, n1, . . . , nl } an1 1 · · · anl l z l∑ i=1 ini , where the sum takes over n0, . . . , nm ≥ 0 and n0 = n1 + · · ·+ nl. Proof. Denote by w the polynomial ∑l i=1 aiz i. For any integer h, we have (1 + w)h = ∑ n0≥0 ( h n0 ) wn0 . Then replacing w with ∑l i=1 aiz i, we obtain( 1 + l∑ i=1 aiz i )h = ∑( h n0 )( n0 n1, . . . , nl ) an1 1 · · · anl l z l∑ i=1 ini . ■ Lemma 4.11. Given integers h1, . . . , hl and the same setup as Theorem 4.7, k∏ j=1 L hj j = ∑ k∏ j=1   hj + k∑ l=j+1 (r(k)∑ s=1 snls ( ĉ+l ,−d(j)cj ) D0R ) nj0, n j 1, . . . , n j r(j)  r(j)∏ s=1 zn j s ij ,s  ŷ k∑ j=1 ( r(j)∑ s=1 snj s)c + j , where the sum takes over all non-negative integers n10, n 1 1, . . . , n 1 r(1) , . . . , nk0, n k 1, . . . , n k r(k) ∈ Z≥0. Proof. We prove the following claim by induction: for all 1 ≤ p ≤ k, k∏ j=p L hj j = ∑ np 0,n p 1,...,n p r(p) ;...;nk 0 ,n k 1 ,...,n k r(k) ∈Z≥0 p−1∏ j=1 L ( k∑ l=p r(l)∑ s=1 snl s(ĉ + l ,−d(j)cj)D0R ) j × k∏ j=p   hj + k∑ l=j+1 ( r(l)∑ s=1 snls(ĉ + l ,−d(j)cj)D0R ) nj0, n j 1, . . . , n j r(j)  r(j)∏ s=1 zn j s ij ,s  ŷ k∑ j=p ( r(j)∑ s=1 snj s)c + j . (4.10) For any integer h and 1 ≤ l ≤ k, by Lemma 4.10, we have Lh l =  r(l)∑ s=0 zil,s ŷc+l l−1∏ j=1 L (ĉ+l ,−d(j)cj)D0R j sh = ∑ n0,n1,...,nr(l) ∈Z≥0 ({ h n0, n1, . . . , nr(l) } r(l)∏ s=1 ( zil,sŷ sc+l l−1∏ j=1 L s(ĉ+l ,−d(j)cj)D0R j )ns ) On F -Polynomials for Generalized Quantum Cluster Algebras and Gupta’s Formula 25 = ∑ n0,n1,...,nr(l) ∈Z≥0 l−1∏ j=1 L r(l)∑ s=1 sns(ĉ + l ,−d(j)cj)D0R j { h n0, n1, . . . , nr(l) }(r(l)∏ s=1 zns il,s ) ŷ r(l)∑ s=1 snsc + l . By taking l = k and h = hk, this prove (4.10) for p = k. Now suppose that (4.10) is true for p+ 1 , then multiply both sides by L hp p , we get k∏ j=p L hj j = L hp p k∏ j=p+1 L hj j = ∑ np+1 0 ,np+1 1 ,...,np+1 r(p+1) ;...;nk 0 ,n k 1 ,...,n k r(k) ∈Z≥0 p−1∏ j=1 L ( k∑ l=p+1 r(l)∑ s=1 snl s(ĉ + (l) ,−d(j)cj)D0R ) j × L hp+ ( k∑ l=p+1 r(l)∑ s=1 snl s(ĉ + (l) ,−d(p)cp)D0R ) p × k∏ j=p+1   hj + k∑ l=j+1 ( r(l)∑ s=1 snls(ĉ + l ,−d(j)cj)D0R ) nj0, n j 1, . . . , n j r(j)  r(j)∏ s=1 zn j s ij ,s  ŷ k∑ j=p+1 (r(j)∑ s=1 snj s ) c+j = ∑ np 0,n p 1,...,n p r(p) ;...;nk 0 ,n k 1 ,...,n k r(k) ∈Z≥0 p−1∏ j=1 L ( k∑ l=p r(l)∑ s=1 snl s(ĉ + l ,−d(j)cj)D0R ) j × k∏ j=p   hj + k∑ l=j+1 ( r(l)∑ s=1 snls ( ĉ+l ,−d(j)cj ) D0R ) nj0, n j 1, . . . , n j r(j)  r(j)∏ s=1 zn j s ij ,s  ŷ k∑ j=p (r(j)∑ s=1 snj s ) c+j . ■ By applying Lemma 4.11 and Theorem 4.7, we have the alternative sum version of Gupta’s formula (4.8). Theorem 4.12. Under the assumption of Theorem 4.7, we have Fik;tk(ŷ, z) = ∑ (n1 0,n 1 1,...,n 1 r(1) ;...;nk 0 ,n k 1 ,...,n k r(k) )∈Z≥0 k∏ j=1 ( Aj r(j)∏ s=1 zn j s ij ,s ) ŷ ∑k j=1( ∑r(j) s=1 snj s)c + j , (4.11) where Aj =  −(gk, d(j)cj)D0R + k∑ l=j+1 ( r(l)∑ s=1 snls(ĉ + l ,−d(j)cj)D0R ) nj0, n j 1, . . . , n j rj  . Remark 4.13. If r = (1, . . . , 1), then formulas (4.8) and (4.11) specialize to [17, Theorems 3.1 and 6.2]. Acknowledgements The authors are grateful to Professor Xueqing Chen for helpful comments. The authors thank the referees for their valuable comments and suggestions in making this article more readable. This work is partially supported by the National Natural Science Foundation of China (Grant No. 11971326). 26 C. Fu, L. Peng and H. Ye References [1] Bai L., Chen X., Ding M., Xu F., A quantum analog of generalized cluster algebras, Algebr. Represent. Theory 21 (2018), 1203–1217, arXiv:1610.09803. 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id	nasplib_isofts_kiev_ua-123456789-212340
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn	1815-0659
language	English
last_indexed	2026-03-21T09:25:29Z
publishDate	2024
publisher	Інститут математики НАН України
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spelling	Fu, Changjian Peng, Liangang Ye, Huihui 2026-02-05T09:53:24Z 2024 On -Polynomials for Generalized Quantum Cluster Algebras and Gupta's Formula. Changjian Fu, Liangang Peng and Huihui Ye. SIGMA 20 (2024), 080, 26 pages 1815-0659 2020 Mathematics Subject Classification: 13F60; 16S34; 05E16 arXiv:2401.15601 https://nasplib.isofts.kiev.ua/handle/123456789/212340 https://doi.org/10.3842/SIGMA.2024.080 We show the polynomial property of -polynomials for generalized quantum cluster algebras and obtain the associated separation formulas under a mild condition. Along the way, we obtain Gupta's formulas of -polynomials for generalized quantum cluster algebras. These formulas specialize to Gupta's formulas for quantum cluster algebras and cluster algebras, respectively. Finally, a generalization of Gupta's formula has also been discussed in the setting of generalized cluster algebras. The authors are grateful to Professor Xueqing Chen for helpful comments. The authors thank the referees for their valuable comments and suggestions in making this article more readable. This work is partially supported by the National Natural Science Foundation of China (Grant No. 11971326). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On -Polynomials for Generalized Quantum Cluster Algebras and Gupta's Formula Article published earlier
spellingShingle	On -Polynomials for Generalized Quantum Cluster Algebras and Gupta's Formula Fu, Changjian Peng, Liangang Ye, Huihui
title	On -Polynomials for Generalized Quantum Cluster Algebras and Gupta's Formula
title_full	On -Polynomials for Generalized Quantum Cluster Algebras and Gupta's Formula
title_fullStr	On -Polynomials for Generalized Quantum Cluster Algebras and Gupta's Formula
title_full_unstemmed	On -Polynomials for Generalized Quantum Cluster Algebras and Gupta's Formula
title_short	On -Polynomials for Generalized Quantum Cluster Algebras and Gupta's Formula
title_sort	on -polynomials for generalized quantum cluster algebras and gupta's formula
url	https://nasplib.isofts.kiev.ua/handle/123456789/212340
work_keys_str_mv	AT fuchangjian onpolynomialsforgeneralizedquantumclusteralgebrasandguptasformula AT pengliangang onpolynomialsforgeneralizedquantumclusteralgebrasandguptasformula AT yehuihui onpolynomialsforgeneralizedquantumclusteralgebrasandguptasformula

On -Polynomials for Generalized Quantum Cluster Algebras and Gupta's Formula

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