Monomial Crystals and Partition Crystals

Recently Fayers introduced a large family of combinatorial realizations of the fundamental crystal for affine sl(n), 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.


Introduction
Fix n ≥ 3 and let B(Λ 0 ) be the crystal corresponding to the fundamental representation of sl n . Recently Fayers [2] constructed an uncountable family of combinatorial realizations of B(Λ 0 ), 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]). Nakajima 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(Λ 0 ) for sl n , 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(Λ 0 ) using Nakajima's monomial crystal and one case of Fayers' partition crystal. Of course any two realizations of B(Λ 0 ) 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.

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 sl n . For us, an sl n crystal is the crystal associated to an integrable highest weight sl n module. It consists of a set B along with operatorsẽī,fī : B → B ∪ {0} for eachī 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 theẽī andfī, so we will not count them as part of the data.

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 sl n when n is even. However, in Section 5 we show that for the fundamental crystal B(Λ 0 ) 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 I be the set of pairs (ī, k) whereī is a residue mod n and k ∈ Z.
• Let M be the set of monomials in the variables Y ±1 ı,k . To be precise, a monomial m ∈ M is a product (ī,k)∈ I Y uī ,k ı,k with all uī ,k ∈ Z and uī ,k = 0 for all but finitely many (ī, k) ∈ I. (1)    We calculate e M 1 and f M 1 . The factors Yī ,k of m are arranged from left to right by decreasing k. The string of brackets S1(m) is as shown above the monomial. The first uncanceled ")" from the right corresponds to a factor of Y −1 1,13 .
We find it convenient to use the following slightly different but equivalent definition ofẽ M ı andf M ı . For eachī modulo n, let Sī(m) be the string of brackets which contains a "(" for every factor of Yī ,k in m and a ")" for every factor of Y −1 ı,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 ı,k+1 m if the first uncanceled "(" from the left comes from a factor Yī ,k . ( It is a straightforward exercise to see that the operators defined in (2) agree with those in (1).

Fayers' crystal structures 4.1 Partitions
A partition λ is a finite length non-increasing sequence of positive integers. Associated to a partition is its Ferrers diagram. We draw these diagrams as in Fig. 2 so that, if λ = (λ 1 , . . . , λ N ), then λ i 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 (x b , y b ) of the center of b, using the axes shown in Fig. 2 is the length of the row containing b and λ ′ y b +1/2 is the length of the column containing b.

The general construction
We now recall Fayers' construction [2] of the crystal B Λ 0 for sl n in its most general version. We begin with some notation. An arm sequence is a sequence A = A 1 , A 2 , . . . of integers such that 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) = A t . A partition λ is called A-regular if it has no A-illegal boxes. Let B A denote the set of A-regular partitions. Fix λ ∈ P and a residueī 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.
For each partition λ and each arm sequence A, define a total order ≻ A on Aī(λ) ∪ Rī(λ) as follows. Fix b = (x, y), b ′ = (x ′ , y ′ ) ∈ Aī(λ) ∪ Rī(λ), and assume b = b ′ . Then there is some It follows from the definition of an arm sequence that ≻ A is transitive.
Fix a partition λ. Construct a string of brackets S Ā ı (λ) by placing a "(" for every b ∈ Aī(λ) and a ")" for every b ∈ Rī(λ), in decreasing order from left to right according to ≻ A . Cancel brackets according to the usual rule. Define mapsẽ Ā ı ,f Ā ı : if there is no uncanceled "(" in S Ā ı . (3)

Special case: the horizontal crystal
Consider the case of the construction given in Section 4.2 where, for all t, A t = ⌈nt/2⌉ − 1 (it is straightforward to see that this satisfies the definition of an arm sequence    Fix λ ∈ B H , and let where The case t < 0 follows immediately from the case t > 0 since ≻ A H is a total order. Lemma 4.4 implies thatẽ H ı andf H ı are as described as in Fig. 3.

A crystal isomorphism
Here A(λ) and R(λ) are as in Section 4.2.
Theorem 5.1. For any n ≥ 3, anyī modulo n, and any λ ∈ B H , we have Before proving Theorem 5.1 we will need a few technical lemmas.
• b ∈ Rī(µ)\Rī(λ). We demonstrate the calculation off2(λ). The string of brackets S H 2 (λ) has a "(" for each2-addable box and a ")" for each2-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). Thusf2(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 Y3 ,11 11 . After reordering and simplifying, this becomes The string of brackets S M 2 (Ψ(λ)) is the same as the string of brackets S H 2 (λ), except that a canceling pair () has been removed. The condition that λ has no A H -illegal boxes implies that S H ı (λ) and S M ı (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): Aī +1 (µ)\Aī +1 (λ) = b ′ and Rī +1 (λ) = Rī +1 (µ) for some box • Either (i): By the definition of Ψ, this implies Proof . By the definitions of Aī(λ) and Rī(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 A H -illegal (see Fig. 4). To see part (iii), recall that by   (λ) and b ′ ∈ Rī(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 ∈ A0(λ) and b ′ ∈ R0(λ). Then hook(a) = 9 = 3 × 3 and arm(a) = 4 = A H 3 , so a is A H -illegal. It is straightforward to see that in general, if either (i):

Part (iv) and (v) follow because any other
Proof of Theorem 5.1. Fix λ ∈ B H andī ∈ Z/nZ. Let S M ı (m) denote the string of brackets used in Section 3 to calculateẽ M ı (m) andf M ı (m). Let S H ı (λ) denote the string of brackets used in Section 4 to calculateẽ H ı (λ) andf H ı (λ), and define the height of a bracket in S H ı (λ) to be h(b) for the corresponding box b ∈ Aī(λ) ∪ Rī(λ).
By Lemma 5.3 parts (iv) and (v), for each k ≥ 1, all "(" in S H ı (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 S H ı (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 S H ı (λ) to calculateẽ H ı (λ) andf H ı (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 Yī ,k in Ψ(λ).
(ii) The ")" in T of height k − 1 correspond exactly to the factors of Y −1 ı,k in Ψ(λ).
Thus the brackets in T correspond exactly to the brackets in S M ı (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)). Proof . This follows immediately from Theorem 5.1, since, by Theorem 4.1, B H is a copy of the crystal B(Λ 0 ).

Questions
Question 1. Nakajima originally developed the monomial crystal using the theory of q-characters 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 Kashiwara [10]. She works with more general integral highest weight crystals, but restricting her results to B(Λ 0 ) one finds a natural isomorphisms between this modified monomial crystal and 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 sl n crystals. There are also natural realizations of higher level sl n 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?