Monomial Crystals and Partition Crystals

Recently Fayers introduced a large family of combinatorial realizations of the fundamental crystal B(Λ₀) for sln, where the vertices are indexed by certain partitions. He showed that special cases of this construction agree with the Misra-Miwa realization and with Berg's ladder crystal. Here we...

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Published in:	Symmetry, Integrability and Geometry: Methods and Applications
Date:	2010
Main Author:	Tingley, P.
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Language:	English
Published:	Інститут математики НАН України 2010
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Cite this:	Monomial Crystals and Partition Crystals / P. Tingley // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 14 назв. — англ.

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spelling	Tingley, P. 2019-02-09T09:26:02Z 2019-02-09T09:26:02Z 2010 Monomial Crystals and Partition Crystals / P. Tingley // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 14 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 17B37; 05E10 DOI:10.3842/SIGMA.2010.035 https://nasplib.isofts.kiev.ua/handle/123456789/146353 Recently Fayers introduced a large family of combinatorial realizations of the fundamental crystal B(Λ₀) for sln, where the vertices are indexed by certain partitions. He showed that special cases of this construction agree with the Misra-Miwa realization and with Berg's ladder crystal. Here we show that another special case is naturally isomorphic to a realization using Nakajima's monomial crystal. We thank Chris Berg, Matthew Fayers, David Hernandez and Monica Vazirani for interesting discussions. This work was supported by NSF grant DMS-0902649. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Monomial Crystals and Partition Crystals Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Monomial Crystals and Partition Crystals
spellingShingle	Monomial Crystals and Partition Crystals Tingley, P.
title_short	Monomial Crystals and Partition Crystals
title_full	Monomial Crystals and Partition Crystals
title_fullStr	Monomial Crystals and Partition Crystals
title_full_unstemmed	Monomial Crystals and Partition Crystals
title_sort	monomial crystals and partition crystals
author	Tingley, P.
author_facet	Tingley, P.
publishDate	2010
language	English
container_title	Symmetry, Integrability and Geometry: Methods and Applications
publisher	Інститут математики НАН України
format	Article
description	Recently Fayers introduced a large family of combinatorial realizations of the fundamental crystal B(Λ₀) for sln, where the vertices are indexed by certain partitions. He showed that special cases of this construction agree with the Misra-Miwa realization and with Berg's ladder crystal. Here we show that another special case is naturally isomorphic to a realization using Nakajima's monomial crystal.
issn	1815-0659
url	https://nasplib.isofts.kiev.ua/handle/123456789/146353
citation_txt	Monomial Crystals and Partition Crystals / P. Tingley // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 14 назв. — англ.
work_keys_str_mv	AT tingleyp monomialcrystalsandpartitioncrystals
first_indexed	2025-11-26T17:52:19Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 6 (2010), 035, 8 pages Monomial Crystals and Partition Crystals Peter TINGLEY Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA E-mail: ptingley@math.mit.edu URL: http://www-math.mit.edu/∼ptingley/ Received February 10, 2010, in final form April 12, 2010; Published online April 21, 2010 doi:10.3842/SIGMA.2010.035 Abstract. Recently Fayers introduced a large family of combinatorial realizations of the fundamental crystal B(Λ0) for ŝln, where the vertices are indexed by certain partitions. He showed that special cases of this construction agree with the Misra–Miwa realization and with Berg’s ladder crystal. Here we show that another special case is naturally isomorphic to a realization using Nakajima’s monomial crystal. Key words: crystal basis; partition; affine Kac–Moody algebra 2010 Mathematics Subject Classification: 17B37; 05E10 1 Introduction Fix n ≥ 3 and let B(Λ0) be the crystal corresponding to the fundamental representation of ŝln. Recently Fayers [2] constructed an uncountable family of combinatorial realizations of B(Λ0), all of whose underlying sets are indexed by certain partitions. Most of these are new, although two special cases have previously been studied. One is the well known Misra–Miwa realization [12]. The other is the ladder crystal developed by Berg [1]. The monomial crystal was introduced by Nakajima in [13, Section 3] (see also [5, 10]). Nakaji- ma considers a symmetrizable Kac–Moody algebra whose Dynkin diagram has no odd cycles, and constructs combinatorial realizations for the crystals of all integrable highest weight modules. In the case of the fundamental crystal B(Λ0) for ŝln, we shall see that the construction works exactly as stated in all cases, including n odd when there is an odd cycle. Here we construct an isomorphism between a realization of B(Λ0) using Nakajima’s monomial crystal and one case of Fayers’ partition crystal. Of course any two realizations of B(Λ0) are isomorphic, so the purpose is not to show that the two realizations are isomorphic, but rather to give a simple and natural description of that isomorphism. This article is organized as follows. Sections 2, 3 and 4 review necessary background material. Section 5 contains the statement and proof of our main result. In Section 6 we briefly discuss some questions arising from this work. 2 Crystals In Sections 3 and 4 we review the construction of Nakajima’s monomial crystals and Fayers’ partition crystals. We will not assume the reader has any prior knowledge of these constructions. We will however assume that the reader is familiar with the notion of a crystal, so will only provide enough of an introduction to that subject to fix conventions, and refer the reader to [9] or [6] for more details. We only consider crystals for the affine Kac–Moody algebra ŝln. For us, an ŝln crystal is the crystal associated to an integrable highest weight ŝln module. It consists of a set B along with mailto:ptingley@math.mit.edu http://www-math.mit.edu/~ptingley/ http://dx.doi.org/10.3842/SIGMA.2010.035 2 P. Tingley operators ẽı̄, f̃ı̄ : B → B ∪ {0} for each ı̄ modulo n, which satisfy various axioms. Often the definition of a crystal includes three functions wt, ϕ, ε : B → P , where P is the weight lattice. In the case of crystals of integrable modules, these functions can be recovered (up shifting in a null direction) from the knowledge of the ẽı̄ and f̃ı̄, so we will not count them as part of the data. 3 The monomial crystal This construction was first introduced in [13, Section 3], where it is presented for symmetrizable Kac–Moody algebras where the Dynkin diagram has no odd cycles. In particular, it only works for ŝln when n is even. However, in Section 5 we show that for the fundamental crystal B(Λ0) the most naive generalization to the case of odd n gives rise to the desired crystal, so the results in this note hold for all n ≥ 3. We now fix some notation, largely following [13, Section 3]. • Let Ĩ be the set of pairs (̄ı, k) where ı̄ is a residue mod n and k ∈ Z. • Define commuting formal variables Yı̄,k for all pairs (̄ı, k) ∈ Ĩ. • Let M be the set of monomials in the variables Y ±1 ı̄,k . To be precise, a monomial m ∈ M is a product ∏ (ı̄,k)∈Ĩ Y uı̄,k ı̄,k with all uı̄,k ∈ Z and uı̄,k = 0 for all but finitely many (̄ı, k) ∈ Ĩ. • For each pair (̄ı, k) ∈ Ĩ, let Aı̄,k = Yı̄,k−1Yı̄,k+1Y −1 ı̄+1̄,k Y −1 ı̄−1̄,k . • Fix a monomial m = ∏ (ı̄,n)∈Ĩ Y uı̄,k ı̄,k ∈M. For L ∈ Z and ı̄ ∈ Z/nZ, define: wt(m) := ∑ (ı̄,k)∈Ĩ uı̄,kΛı̄, εı̄,L(m) := − ∑ l≥L uı̄,l(m), ϕı̄,L(m) := ∑ l≤L uı̄,l(m), εı̄(m) := max{εı̄,L(m) \| L ∈ Z}, pı̄(m) := max{L ∈ Z \| εı̄,L(m) = εı̄(m)}, ϕı̄(m) := max{ϕı̄,L(m) \| L ∈ Z}, qı̄(m) := min{L ∈ Z \| ϕı̄,L(m) = ϕı̄(m)}. Note that one always has ϕı̄(m), εı̄(m) ≥ 0. Furthermore, if εı̄(m) > 0 then pı̄ is finite, and if ϕı̄(m) > 0 then qı̄(m) is finite. • Define ẽM ı̄ , f̃M ı̄ : M∪{0} →M∪ {0} for each residue ı̄ modulo n by ẽM ı̄ (0) = f̃M ı̄ (0) = 0, ẽM ı̄ (m) := { 0 if εı̄(m) = 0, Aı̄,pı̄(m)−1m if εı̄(m) > 0, f̃M ı̄ (m) := { 0 if ϕı̄(m) = 0, A−1 ı̄,qı̄(m)+1m if ϕı̄(m) > 0. (1) Definition 3.1. A monomial m is called dominant if uı̄,k ≥ 0 for all (̄ı, k) ∈ Ĩ . Definition 3.2. Assume n is even. Then m is called compatible if uı̄,k 6= 0 implies k ∼= ı̄ modulo 2. Definition 3.3. Let M(m) be the set of monomials in M which can be reached from m by applying various ẽM ı̄ and f̃M ı̄ . Theorem 3.4 ([13, Theorem 3.1]). Assume n > 2 is even, and let m be a dominant, compa- tible monomial. Then M(m) along with the operators ẽM ı̄ and f̃M ı̄ is isomorphic to the crystal B(wt(m)) of the integrable highest weight ŝln module V (wt(m)). Remark 3.5. Notice that although Theorem 3.4 only holds when n is even, the operators ẽM ı̄ and f̃M ı̄ are well defined for any n ≥ 3 and any monomial m. When n is odd or m does not satisfy the conditions of Theorem 3.4, M(m) need not be a crystal. However, as we prove in Section 5, even when n is odd M(Y0̄,0) is a copy of the crystal B(Λ0) of the fundamental representation of ŝln. Monomial Crystals and Partition Crystals 3 Figure 1. The operators ẽM ı̄ and f̃M ı̄ on a monomial m ∈ M for n = 4. We calculate ẽM 1̄ and f̃M 1̄ . The factors Yı̄,k of m are arranged from left to right by decreasing k. The string of brackets S1̄(m) is as shown above the monomial. The first uncanceled “)” from the right corresponds to a factor of Y −1 1̄,13 . Thus ẽM 1̄ (m) = A1̄,12m = Y1̄,11Y1̄,13Y −1 0̄,12 Y −1 2̄,12 m. The first uncanceled “(” from the left corresponds to a factor of Y1̄,9, so f̃M 1̄ (m) = A−1 1̄,10 m = Y −1 1̄,9 Y −1 1̄,11 Y0̄,10Y2̄,10m. We find it convenient to use the following slightly different but equivalent definition of ẽM ı̄ and f̃M ı̄ . For each ı̄ modulo n, let Sı̄(m) be the string of brackets which contains a “(” for every factor of Yı̄,k in m and a “)” for every factor of Y −1 ı̄,k ∈ m, for all k ∈ Z. These are ordered from left to right in decreasing order of k, as shown in Fig. 1. Cancel brackets according to usual conventions, and set ẽM ı̄ (m)= { 0 if there is no uncanceled “)” in m, Aı̄,k−1m if the first uncanceled “)” from the right comes from a factor Y −1 ı̄,k , f̃M ı̄ (m)= { 0 if there is no uncanceled “(” in m, A−1 ı̄,k+1m if the first uncanceled “(” from the left comes from a factor Yı̄,k. (2) It is a straightforward exercise to see that the operators defined in (2) agree with those in (1). 4 Fayers’ crystal structures 4.1 Partitions A partition λ is a finite length non-increasing sequence of positive integers. Associated to a par- tition is its Ferrers diagram. We draw these diagrams as in Fig. 2 so that, if λ = (λ1, . . . , λN ), then λi is the number of boxes in row i (rows run southeast to northwest). Let P denote the set of all partitions. For λ, µ ∈ P, we say λ is contained in µ if the diagram for λ fits inside the diagram for µ. Fix λ ∈ P and a box b in (the diagram of) λ. We now define several statistics of b. See Fig. 2 for an example illustrating these. The coordinates of b are the coordinates (xb, yb) of the center of b, using the axes shown in Fig. 2. The content c(b) is yb − xb. The arm length of b is arm(b) := λxb+1/2− yb− 1/2, where λxb+1/2 is the length of the row through b. The hook length of b is hook(b) := λxb+1/2− yb +λ′yb+1/2−xb, where λxb+1/2 is the length of the row containing b and λ′yb+1/2 is the length of the column containing b. 4.2 The general construction We now recall Fayers’ construction [2] of the crystal BΛ0 for ŝln in its most general version. We begin with some notation. An arm sequence is a sequence A = A1, A2, . . . of integers such that (i) t− 1 ≤ At ≤ (n− 1)t for all t ≥ 1, and (ii) At+u ∈ [At + Au, At + Au + 1] for all t, u ≥ 1. Fix an arm sequence A. A box b in a partition λ is called A-illegal if, for some t ∈ Z>0, hook(b) = nt and arm(b) = At. A partition λ is called A-regular if it has no A-illegal boxes. Let BA denote the set of A-regular partitions. 4 P. Tingley Figure 2. The partition λ = (7, 6, 5, 5, 5, 3, 3, 1), drawn in “Russian” notation. The parts λi are the lengths of the “rows” of boxes sloping southeast to northwest. The center of each box b has coordinate (xb, yb) for some xb, yb ∈ Z+1/2. For the box labeled a, xa = 2.5 and ya = 1.5. The content c(a) = ya−xa records the horizontal position of a, reading right to left. In this case, c(a) = −1. Other relevant statistics are hook(a) = 8, arm(a) = 3 and h(a) = 4. This partition is not in BH for ŝl4, since the box a is AH - illegal. For λ ∈ P and a box b ∈ λ. The color of b is the residue c(b) of c(b) modulo n, where as in Section 4.1, c(b) is the content of b. See Fig. 3. Fix λ ∈ P and a residue ı̄ modulo n. Define • A(λ) is the set of boxes b which can be added to λ so that the result is still a partition. • R(λ) is the set of boxes b which can be removed from λ so that the result is still a partition. • Aı̄(λ) = {b ∈ A(λ) such that c(b) = ı̄}. • Rı̄(λ) = {b ∈ R(λ) such that c(b) = ı̄}. For each partition λ and each arm sequence A, define a total order �A on Aı̄(λ) ∪ Rı̄(λ) as follows. Fix b = (x, y), b′ = (x′, y′) ∈ Aı̄(λ) ∪ Rı̄(λ), and assume b 6= b′. Then there is some t ∈ Z\{0} such c(b′) − c(b) = nt. Interchanging b and b′ if necessary, assume t > 0. Define b′ �A b if y′ − y > At, and b �A b′ otherwise. It follows from the definition of an arm sequence that �A is transitive. Fix a partition λ. Construct a string of brackets SA ı̄ (λ) by placing a “(” for every b ∈ Aı̄(λ) and a “)” for every b ∈ Rı̄(λ), in decreasing order from left to right according to �A. Cancel brackets according to the usual rule. Define maps ẽA ı̄ , f̃A ı̄ : P ∪ {0} → P ∪ {0} by eA ı̄ (0) = fA ı̄ (0) = 0, ẽA ı̄ (λ) = { λ\b if the first uncanceled “)” from the right in SA ı̄ corresponds to b, 0 if there is no uncanceled “)” in SA ı̄ , f̃A ı̄ (λ) = { λ t b if the first uncanceled “(” from the left in SA ı̄ corresponds to b, 0 if there is no uncanceled “(” in SA ı̄ . (3) Theorem 4.1 ([2, Theorem 2.2]). Fix n ≥ 3 and an arm sequence A. Then BA ∪ {0} is preserved by the maps ẽA ı̄ and f̃A ı̄ , and forms a copy of the crystal B(Λ0) for ŝln, where as above BA is the set of partitions with no A-illegal hooks. Remark 4.2. The operators ẽı̄ and f̃ı̄ are defined on all partitions. However, as noted in [2], the component generated by a non A-regular partition need not be an ŝln crystal. Monomial Crystals and Partition Crystals 5 4.3 Special case: the horizontal crystal Consider the case of the construction given in Section 4.2 where, for all t, At = dnt/2e − 1 (it is straightforward to see that this satisfies the definition of an arm sequence). This arm sequence will be denoted AH . For convenience, we denote BAH simply by BH and the operators ẽAH ı̄ and f̃AH ı̄ from Section 4.2 by ẽH ı̄ and f̃H ı̄ . Definition 4.3. Let b = (x, y) be a box. The height of b is h(b) := x + y. Lemma 4.4. Fix λ ∈ BH , and let b, b′ ∈ Aı̄(λ)∪Rı̄(λ) with b 6= b′. Then b′ �AH b if and only if (i) h(b′) > h(b), or (ii) h(b′) = h(b) and c(b′) > c(b). Proof. Since c̄(b) = c̄(b′), we have c(b′)− c(b) = nt for some t ∈ Z\{0}. First consider the case when t > 0. Then by definition, b′ �AH b if and only if y′ − y > AH t , (4) where AH t = dnt/2e − 1 = d(c(b′)− c(b))/2e − 1 = d(y′ − x′ − y + x)/2e − 1 = { (y′ − x′ − y + x)/2− 1 if y′ − x′ − y + x is even, (y′ − x′ − y + x)/2− 1/2 if y′ − x′ − y + x is odd. Since y′ − y ∈ Z, (4) holds if and only if y′ − y > (y′ − x′ − y + x)/2− 1/2, which rearranges to h(b′) ≥ h(b). The case t < 0 follows immediately from the case t > 0 since �AH is a total order. � Lemma 4.4 implies that ẽH ı̄ and f̃H ı̄ are as described as in Fig. 3. 5 A crystal isomorphism Define a map Ψ : P ∪ {0} →M∪ {0} by Ψ(0) = 0, and, for all λ ∈ P, Ψ(λ) := ∏ b∈A(λ) Yc̄(b),h(b)−1 ∏ b∈R(λ) Y −1 c̄(b),h(b)+1. Here A(λ) and R(λ) are as in Section 4.2. Theorem 5.1. For any n ≥ 3, any ı̄ modulo n, and any λ ∈ BH , we have Ψ(ẽH ı̄ λ) = ẽM ı̄ Ψ(λ) and Ψ(f̃H ı̄ λ) = f̃M ı̄ Ψ(λ), where ẽM ı̄ , f̃M ı̄ are as in Section 3 and ẽH ı̄ , f̃H ı̄ are as in Section 4.3. Before proving Theorem 5.1 we will need a few technical lemmas. Lemma 5.2. Let λ and µ be partitions such that µ = λ t b for some box b. Then Ψ(µ) = A−1 c(b),h(b)Ψ(λ). Proof. Let i = c̄(b). It is clear that the pair (A(λ), R(λ)) differs from (A(µ), R(µ)) in exactly the following four ways: • b ∈ Aı̄(λ)\Aı̄(µ). • b ∈ Rı̄(µ)\Rı̄(λ). 6 P. Tingley Figure 3. The partition λ = (11, 7, 4, 2, 1, 1, 1, 1, 1, 1) is an element of BH for ŝl4, since no box b in λ is AH -illegal. The color c(b) of each box b is written inside b. We demonstrate the calculation of f̃2̄(λ). The string of brackets SH 2̄ (λ) has a “(” for each 2̄-addable box and a “)” for each 2̄-removable box, ordered from left to right lexicographically, first by decreasing height h(b), then by decreasing content c(b). The result is shown on the right of the diagram (rotate the page 90 degrees counter clockwise). Thus f̃2̄(λ) is the partition obtained by adding the box with coordinates (x, y) = (10.5, 0.5). The map Ψ from Section 5 takes λ to Y3̄,11Y2̄,8Y2̄,6Y3̄,5Y1̄,5Y2̄,10Y −1 2̄,12 Y −1 1̄,9 Y −1 1̄,7 Y −1 2̄,6 Y −1 3̄,11 . After reordering and simplifying, this becomes Y −1 2̄,12 Y2̄,10Y −1 1̄,9 Y2̄,8Y −1 1̄,7 Y1̄,5Y3̄,5. The string of brackets SM 2̄ (Ψ(λ)) is the same as the string of brackets SH 2̄ (λ), except that a canceling pair () has been removed. The condition that λ has no AH -illegal boxes implies that SH ı̄ (λ) and SM ı̄ (Ψ(λ)) are always the same up to removing pairs of canceling brackets, which is essentially the proof that Ψ is an isomorphism. • Either (i): Aı̄+1̄(µ)\Aı̄+1̄(λ) = b′ and Rı̄+1̄(λ) = Rı̄+1̄(µ) for some box b′ with h(b′) = h(b) + 1, or (ii): Rı̄+1̄(λ)\Rı̄+1̄(µ) = b′ and Aı̄+1̄(λ) = Aı̄+1̄(µ) for some box b′ with h(b′) = h(b)− 1. • Either (i): Aı̄−1̄(µ)\Aı̄−1̄(λ) = b′′ and Rı̄−1̄(λ) = Rı̄−1̄(µ) for some box b′′ with h(b′′) = h(b) + 1, or (ii): Rı̄−1̄(λ)\Rı̄−1̄(µ) = b′′ and Aı̄−1̄(λ) = Aı̄−1̄(µ) for some box b′′ with h(b′′) = h(b)− 1. By the definition of Ψ, this implies Ψ(µ) = Y −1 ı̄,h(b)−1Y −1 ı̄,h(b)+1Yı̄+1̄,h(b)Yı̄−1̄,h(b)Ψ(λ) = A−1 ı̄,h(b)Ψ(λ). � Lemma 5.3. Let λ ∈ BH , and choose b ∈ Aı̄(λ), b′ ∈ Rı̄(λ). Then (i) h(b) 6= h(b′) + 1. (ii) If h(b) = h(b′) then c(b′) > c(b), so b′ �AH b. (iii) b �AH b′ if and only if h(b)− 1 ≥ h(b′) + 1. (iv) Assume h(b)− 1 = h(b′) + 1. If c ∈ Aı̄(λ) satisfies b �AH c �AH b′, then h(c) = h(b). (v) Assume h(b)− 1 = h(b′) + 1. If c ∈ Rı̄(λ) satisfies b �AH c �AH b′, then h(c) = h(b′). Proof. By the definitions of Aı̄(λ) and Rı̄(λ), b and b′ cannot lie in either the same row or the same column, which implies that there is a unique box a in λ which shares a row or column with each of b, b′. It is straightforward to see that if (i) or (ii) is violated then this a is AH -illegal (see Fig. 4). To see part (iii), recall that by Lemma 4.4, b �AH b′ if and only if h(b) > h(b′) or both h(b) = h(b′) and c(b) > c(b′). This order agrees with the formula in part (iii), since parts (i) and (ii) eliminate all cases where they would differ. Monomial Crystals and Partition Crystals 7 Figure 4. The hook defined by b ∈ Aı̄(λ) and b′ ∈ Rı̄(λ). There will always be a unique box a in λ which is in either the same row or the same column as b and also in either the same row or the same column as b′. Taking n = 3, we have b ∈ A0̄(λ) and b′ ∈ R0̄(λ). Then hook(a) = 9 = 3× 3 and arm(a) = 4 = AH 3 , so a is AH -illegal. It is straightforward to see that in general, if either (i): h(b) = h(b′)+1 or (ii): h(b) = h(b′) and c(b) > c(b′), then the resulting hook is AH -illegal, and hence λ /∈ BH . Part (iv) and (v) follow because any other c ∈ Aı̄(λ) ∪ Rı̄(λ) with b �AH c �AH b′ would violate either (i) or (ii). � Proof of Theorem 5.1. Fix λ ∈ BH and ı̄ ∈ Z/nZ. Let SM ı̄ (m) denote the string of brackets used in Section 3 to calculate ẽM ı̄ (m) and f̃M ı̄ (m). Let SH ı̄ (λ) denote the string of brackets used in Section 4 to calculate ẽH ı̄ (λ) and f̃H ı̄ (λ), and define the height of a bracket in SH ı̄ (λ) to be h(b) for the corresponding box b ∈ Aı̄(λ) ∪Rı̄(λ). By Lemma 5.3 parts (iv) and (v), for each k ≥ 1, all “(” in SH ı̄ (λ) of height k + 1 are immediately to the left of all “)” of height k − 1. Let T be the string of brackets obtained from SH ı̄ (λ) by, for each k, canceling as many “(” of height k + 1 with “)” of height k− 1 as possible. Notice that one can use T instead of SH ı̄ (λ) to calculate ẽH ı̄ (λ) and f̃H ı̄ (λ) without changing the result. By the definition of Ψ, it is clear that (i) The “(” in T of height k + 1 correspond exactly to the factors of Yı̄,k in Ψ(λ). (ii) The “)” in T of height k − 1 correspond exactly to the factors of Y −1 ı̄,k in Ψ(λ). Thus the brackets in T correspond exactly to the brackets in SM ı̄ (Ψ(λ)). Furthermore, Lemma 5.3 part (iii) implies that these brackets occur in the same order. The theorem then follows from Lemma 5.2 and the definitions of the operators (see equations (2) and (3)). � Corollary 5.4. For any n, M(Y0̄,0) is a copy of the fundamental crystal B(Λ(0)) for ŝln. Proof. This follows immediately from Theorem 5.1, since, by Theorem 4.1, BH is a copy of the crystal B(Λ0). � 6 Questions Question 1. Nakajima originally developed the monomial crystal using the theory of q-charac- ters from [4]. Can this theory be modified to give rise to any of Fayers’ other crystal structures? One may hope that this would help explain algebraically why these crystal structures exist. Question 2. In [11], Kim considers a modification to the monomial crystal developed by Kashi- wara [10]. She works with more general integral highest weight crystals, but restricting her results to B(Λ0) one finds a natural isomorphisms between this modified monomial crystal and 8 P. Tingley the Misra–Miwa realization. The Misra–Miwa realization corresponds to one case of Fayer’s partition crystal, but not the one studied in Section 4.3. In [10], there is some choice as to how the monomial crystal is modified. Do other modifications also correspond to instances of Fayers crystal? Which instances of Fayers’ partiton crystal correspond to modified monomial crystals (or appropriate generalizations)? Question 3. The monomial crystal construction works for higher level ŝln crystals. There are also natural realizations of higher level ŝln crystals using tuples of partitions (see [3, 7, 8, 14]). Is there an analogue of Fayers’ construction in higher levels generalizing both of these types of realization? Acknowledgments We thank Chris Berg, Matthew Fayers, David Hernandez and Monica Vazirani for interesting discussions. This work was supported by NSF grant DMS-0902649. References [1] Berg C., (`, 0)-JM partitions and a ladder based model for the basic crystal of ŝl`, arXiv:0901.3565. [2] Fayers M., Partition models for the crystal of the basic Uq(ŝln)-module, J. Algebraic Combin., to appear, arXiv:0906.4129. [3] Foda O.,, Leclerc B., Okado M., Thibon J.-Y., Welsh T.A., Branching functions of A (1) n−1 and Jantzen–Seitz problem for Ariki–Koike algebras, Adv. Math. 141 (1999), 322–365, q-alg/9710007. [4] Frenkel E., Reshetikhin N., The q-characters of representations of quantum affine algebras and deformations of W-algebras, in Recent Developments in Quantum Affine Algebras and Related Topics (Raleigh, NC, 1998), Contemp. Math., Vol. 248, Amer. Math. Soc., Providence, RI, 1999, 163–205, math.QA/9810055. [5] Hernandez D., Nakajima H., Level 0 monomial crystals, Nagoya Math. J. 184 (2006), 85–153, math.QA/0606174. [6] Hong J., Kang S.-J., Introduction to quantum groups and crystal bases, Graduate Studies in Mathematics, Vol. 42, Amer. Math. Soc., Providence, RI, 2002. [7] Jimbo M., Misra K.C., Miwa T., Okado M., Combinatorics of representations of Uq(ŝl(n)) at q = 0, Comm. Math. Phys. 136 (1991), 543–566. [8] Kang S.-J., Lee H., Higher level affine crystals and Young walls, Algebr. Represent. Theory 9 (2006), 593– 632, math.QA/0310430. [9] Kashiwara M., On crystal bases, in Representations of Groups (Banff, AB, 1994), CMS Conf. Proc., Vol. 16, Amer. Math. Soc., Providence, RI, 1995, 155–197. [10] Kashiwara M., Realizations of crystals, in Combinatorial and Geometric Representation Theory (Seoul, 2001), Contemp. Math., Vol. 325, Amer. Math. Soc., Providence, RI, 2003, 133–139, math.QA/0202268. [11] Kim J.-A., Monomial realization of crystal graphs for Uq(An(1)), Math. Ann. 332 (2005), 17–35. [12] Misra K., Miwa T., Crystal base for the basic representation of Uq(ŝln), Comm. Math. Phys. 134 (1990), 79–88. [13] Nakajima H., t-analogs of q-characters of quantum affine algebras of type An, Dn, in Combinatorial and Geometric Representation Theory (Seoul, 2001), Contemp. Math., Vol. 325, Amer. Math. Soc., Providence, RI, 2003, 141–160, math.QA/0204184. [14] Tingley P., Three combinatorial models for ŝln crystals, with applications to cylindric plane partitions, Int. Math. Res. Not. IMRN 2008 (2008), no. 2, Art. ID rnm143, 40 pages, math.QA/0702062. http://arxiv.org/abs/0901.3565 http://dx.doi.org/10.1007/s10801-010-0217-9 http://arxiv.org/abs/0906.4129 http://dx.doi.org/10.1006/aima.1998.1783 http://arxiv.org/abs/q-alg/9710007 http://arxiv.org/abs/math.QA/9810055 http://projecteuclid.org/euclid.nmj/1167159343 http://arxiv.org/abs/math.QA/0606174 http://dx.doi.org/10.1007/BF02099073 http://dx.doi.org/10.1007/BF02099073 http://dx.doi.org/10.1007/s10468-006-9013-6 http://arxiv.org/abs/math.QA/0310430 http://arxiv.org/abs/math.QA/0202268 http://dx.doi.org/10.1007/s00208-004-0613-3 http://dx.doi.org/10.1007/BF02102090 http://arxiv.org/abs/math.QA/0204184 http://dx.doi.org/10.1093/imrn/rnm143 http://dx.doi.org/10.1093/imrn/rnm143 http://arxiv.org/abs/math.QA/0702062 1 Introduction 2 Crystals 3 The monomial crystal 4 Fayers' crystal structures 4.1 Partitions 4.2 The general construction 4.3 Special case: the horizontal crystal 5 A crystal isomorphism 6 Questions References

Monomial Crystals and Partition Crystals

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