Schur Positivity and Kirillov-Reshetikhin Modules

In this note, inspired by the proof of the Kirillov-Reshetikhin conjecture, we consider tensor products of Kirillov-Reshetikhin modules of a fixed node and various level. We fix a positive integer and attach to each of its partitions such a tensor product. We show that there exists an embedding of the tensor products, with respect to the classical structure, along with the reverse dominance relation on the set of partitions.


Introduction
This note is inspired by two results on certain modules of simple, finite-dimensional complex Lie algebras. The first one is an immediate consequence of the famous Clebsch-Gordan formula on decompositions of tensor product of simple sl 2 (C)-modules. Namely let V (m) = Sym m C 2 , be the n-th symmetric power of the natural representation, then Which implies that for m 1 , m 2 ≤ m there exists a surjective map of sl 2 -modules if and only if min{m 1 , m − m 1 } ≤ min{m 2 , m − m 2 }. Using this inequality, we obtain an order on partitions of n of length 2, P(m, 2). By taking the point of view from symmetric functions, s m being the character of V (m), we have (m 1 , m − m 1 ) (m 2 , m − m 2 ) ⇔ s m 2 s m−m 2 − s m 1 s m−m 1 ∈ k≥0 Z ≥0 s k .
As the characters are also known as Schur functions, this property of the left hand side is also known as Schur positivity. A generalization of this order to sl n (C) and further to a simple finite-dimensional Lie algebra g of arbitrary type was investigated in [DP] (resp. [CFS], [F]).
The other inspiration comes from certain character identities for classical limits of Kirillov-Reshetikhin modules for U q (ĝ), the untwisted quantum affine algebra associated to g, namely the Q-systems. Kirillov-Reshetikhin modules W We denote by KR(mω i ) the g-module obtained through the limit q → 1 of W (i) k,a , note that the classical structure is independent of a. Further denote char g KR(mω i ) the classical character of KR(mω i ), then the Q-system is the following character identity ( [H, N]) m denotes the classical limit of a tensor product of certain Kirillov-Reshetikhin modules (depending on i and m). From the Q-system one has the immediate consequence, that there exists a surjective map of g-modules Considering the partial order on partitions, we have (m − 1, m + 1) ≺ (m, m) ∈ P(2m, 2). So for the maximal element and its predecessor this might be seen as a generalization of Schur positivity (to Lie algebras of arbitrary type). Note that in [CV] it is proved that there exists a surjective map of modules for the current algebra of g, namely of fusion products of modules of Kirillov-Reshetikhin modules for the partitions (m − 1, m + 1), (m, m). Their work is also motivated by the Q-system.
Combining the partial order on P(m, 2) and this consequence of the Q-system relation was the starting point of this paper. We have generalized the arguments in the proof of the Kirillov-Reshetikhin conjecture, e.g. the character of the Kirillov-Reshetikhin modules satisfy the Q-system. Using this, we have proved that for all g of arbitrary type and (m 1 , m − m 1 ) (m 2 , m − m 2 ) ∈ P(m, 2) there exists a surjective map of g-modules This might be seen as a generalization of the sl 2 -case as well as of the consequence of the Q-system property.
By considering the reverse dominance relation on partitions of m, P(m), we can show further (using that the cover relation is induced by the cover relations on the set of partitions of length 2) that if (m 1 ≥ . . . ≥ m k 1 > 0) (n 1 ≥ . . . ≥ n k 2 > 0), then there exists a surjective map of g-modules The last statement has been proved in [CFS] for the case where ω i is minuscule, e.g. KR(mω i ) is a simple g-module. The authors were constructing an explicit bijection of the highest weight vectors in terms of LS-paths. Our approach avoids these combinatorics.
Since KR(mω i ) can be also constructed as a module for g ⊗ C[t] (by using some "evaluation parameter" a ∈ C), see Section 1.5, one might ask if there is a surjection also as g ⊗ C[t]-modules. The natural object to be considered here is the fusion product introduced in [FL]. This is the associated graded module (with respect to the degree filtration on U (g ⊗ C[t])) of the tensor product of the Kirillov-Reshetikhin modules with pairwise distinct evaluations. Can the surjection in Theorem 2.1 be actually obtained from a surjective map of the corresponding fusion products?
We should remark here that a similar result on Schur positivity on tensor products of simple g-modules of arbitrary highest weight λ was conjectured in [DP] and [CFS] (see Section 2.4 for more details). Our result suggests, that this generalized Schur positivity may hold along the partial order on tensor products of q → 1 limits of minimal affinizations of V (λ) (the "minimal" module of the quantum affine algebra having a simple quotient whose limit is isomorphic to V (λ), see [CP] for more details). In the sl n -case, the restriction to sl n of the limit of such a minimal affinization is nothing but the simple sl n -module V (λ), so this is the conjecture of Schur positivity by Lam, Postnikov and Pylyavskyy [DP]. For other types, the limit of a minimal affinization is not a simple g-module in general, for example Kirillov-Reshetikhin modules are minimal affinizations of V (mω i ). It might be interesting to investigate on minimal affinizations of other than rectangular weights.
In Section 1 we briefly recall the reverse dominance relation on partitions and some basics on Kirillov-Reshetikhin modules. In Section 2 we state the main theorem, while the proof follows in Section 3.
1.1. Partial order. We recall the reverse dominance order on partitions. For this, let m ≥ 1 be positive integer and denote by P(m) the set of partitions of m: This is a finite set and the reverse dominance relation on P(m) is defined as follows: Obviously, this gives a partial order on P(m) with a smallest element (m > 0) and largest element (1 ≥ 1 ≥ . . . ≥ 1 > 0). Moreover, if we consider partitions of a fixed length k only, P(m, k), then there is also a unique maximal element.
is the unique maximal element in P(m, k).
1.2. We recall the notion of the cover relation induced by , e.g. we say µ covers λ if (i) λ µ and (ii) λ ν µ implies ν = λ or ν = µ Since P(m) is a finite set, we can find for each pair λ µ partitions ν 0 , . . . , ν ℓ such that λ = ν 0 ν 1 . . . ν ℓ = µ and ν i covers ν i−1 for all i. To understand the partial order on P(m) it is therefore sufficient to understand the cover relation on P(m). The following proposition was proved in [CFS,Proposition 3.5] (for simplicity of notation we assume that λ, µ have the same length by adding 0 parts to at most one of the both).
1.3. Quantum affine algebras and their representations. We give a brief reminder on quantum affine algebras and their finite-dimensional representations. For more details we refer to [H, CH].
Let q ∈ C * which is not a root of unity. Let U q (ĝ) be the untwisted quantum affine algebra associated to g. The simple objects of the category C of (type 1) finite-dimensional representations of U q (ĝ) are parametrized by dominant monomials of the ring Y = Z[Y ±1 i,a ] 1≤i≤n,a∈C * , that is for each such monomial m = i∈I,a∈C * Y u i,a i,a which is dominant (the u i,a ≥ 0), there is a corresponding simple object L(m) in C. For example, for i ∈ I, k ≥ 0, a ∈ C * , we have the Kirillov-Reshetikhin module Here q i = q r i where r i is the length of simple root α i .
The q-character morphism [FR] is an injective ring morphism defined on the Grothendieck ring Rep(U q (ĝ)) of the tensor category C: Theorem 1.2. [FR, FM] For m a dominant monomial, we have An element in Im(χ q ) is characterized by the multiplicity of its dominant monomials.
Note that we have a partial ordering on the monomials of Y : m m ′ if m ′ m −1 is a product of monomials A i,a . The first statement in the theorem can be reformulated by saying that all monomials in χ q (L(m)) are lower than m for this ordering, that is m is the highest monomial.
As consequence of the second statement, if we know that the q-character of a simple module has a unique dominant monomial, its q-character can be reconstructed (this is the Frenkel-Mukhin algorithm [FM]). This property has been proved in the important case of Kirillov-Reshetikhin modules, which led to the proof of the Kirillov-Reshetikhin conjecture. It was first proved by Nakajima [N] for ADE-types and in [H] with a different proof which can be extended to the general case.
Theorem 1.3. [N, H] The q-character of Kirillov-Reshetikhin module has a unique dominant monomial. This implies the T -system in the Grothendieck ring Rep(U q (ĝ)) :

a is a tensor product of Kirillov-Reshetikhin modules. The representations S
1.4. The classical limit. Let KR(mω i ) be the restriction of W (i) k,a as a U q (g)-module (it is well known it does not depend on a, see references in [H]). We denote by the same symbol its limit at q = 1 (that is we consider the corresponding g-module).
KR(mω i ) decomposes into a direct sum of finitely many simple g-modules. This decomposition is computed for g of classical type, namely type A, B, C, D as well as for certain nodes for exceptional types in [C2] ([C2, Theorem 1] and [C2,Chapter 3]. For more on decompositions of Kirillov-Reshetikhin modules see [HKOTT].

Chari modules.
For the readers convenience we should clarify the relation to Kirillov-Reshetikhin modules for current algebras, although it is not used in this note at all: Denote g ⊗ C[t] the current algebra of g. In [C2], V. Chari introduced finite-dimensional modules for g ⊗ C[t] which were supposed to be classical analogs of Kirillov-Reshetikhin modules. Namely for m ∈ Z ≥0 , a ∈ C, i ∈ I, C(mω i , a) is the g ⊗ C[t]-module generated by a non-zero vector w subject to the relations a)).w = 0. The following was proven for g of classical type, namely of type A, B, C, D in [C2, Corollary 2.1]. The general case (e.g. an arbitrary simple, finite-dimensional complex Lie algebra) can be deduced as the special case of a single tensor factor of [K,Corollary 5.1], this was proved by using a pentagram of identities see [K,Section 1.2].
Theorem 1.4. For i ∈ I and m ≥ 0 we have C(mω i , a) ∼ = g KR(mω i ).

Main result
2.1. Main theorem. For λ = (m 1 ≥ . . . ≥ m k > 0) ∈ P(m) and i ∈ I we denote the tensor product of the associated KR modules as: Note that KR(0, ω i ) ∼ = C is the trivial representation. We are mainly interested in the gstructure of these modules. Since this is the tensor product of finite-dimensional g-modules, it decomposes into the direct sum of simple modules. Due to the structure of the Kirillov-Reshetikhin modules, we see immediately that the maximal weight occurring is mω i and that the corresponding weight space has dimension 1.
The main purpose is of this paper is the study of the g-structure of these modules along the partial order and the main theorem is the following Theorem 2.1. Let m ≥ 1, i ∈ I and λ µ ∈ P(m), then dim(Hom g (KR(λ, i), V (τ ))) ≤ dim(Hom g (KR(µ, i), V (τ ))) for all τ ∈ P + .
In other words, there exists a surjective map of g-modules Proof. Let λ µ =∈ P(m). We denote µ = (n 1 ≥ . . . ≥ n k 2 ≥ 0). It is sufficient to prove the statement for the case where µ covers λ. In this case, we know by Proposition 1.1 there exists p < q such that n ℓ = m ℓ for all ℓ = p, q. This implies that So to prove the statement it is enough to give a proof for partitions of m of length 2. So let m 1 ≥ m 2 > 0, we have to show that we have a surjective map of g-modules This will be proven in the next section, Theorem 3.1.

2.2.
Remarks. Before proving the last ingredient in the proof (Theorem 3.1), we shall make a couple of remarks. First of all, Theorem 2.1 was proved in [CFS, Theorem ii)] for the special case where ω i is a minuscule weight. In this case, KR(mω i ) ∼ = V (mω i ) as a g-module (see for example [C2]). This case covers the type sl n+1 as well as certain special cases for other types. The proof given there uses the combinatorics of LS paths. Namely, an injection on the level of paths in the tensor product is given. This proof does not extend to other cases since the combinatorics of LS-paths are more complicated for non-minuscule weights.
2.3. The partial order on P(m) is a special case of the more general poset P(λ) introduced in [CFS]. Here λ ∈ P + is a dominant weight and the elements in the set are partitions of λ, namely λ = (λ 1 , . . . , λ k ≥ 0), with λ i ∈ P + and λ 1 + . . . + λ k = λ. Then the partial order considered in the present paper is a generalization of the partial order on P(λ), namely λ µ if and only if for all α ∈ R + and ℓ ≥ 1: Then [CFS,Theorem 1(i) for all τ ∈ P + . This conjecture was made before for g of type sl n+1 by Lam, Postnikov, Pylyavskyy (cited in [DP,Conjecture 1]). In this case, the conjecture is equivalent to the following statement on the level of characters: where s λ is the Schur function corresponding to λ, e.g. the character of the simple sl n+1module V (λ). In other words, it is conjectured that the difference of the products of Schur functions is Schur positive.
Further, in [CFS,Theorem 1 iii] the conjecture was proved for sl 3 .
2.5. Besides some numerical evidence and the partial cases stated above, the conjectures remain open. With the result of the current paper one may tempt to replace the simple modules V (λ) in the tensor product by minimal affinizations of V (λ) (see [CP] for more details on these) and still conjecture a surjective map of g-modules. Note that for classical types, KR(mω i ) is the minimal affinization of V (mω i ) ( [C1,CP]). The proof uses arguments of the proofs of [H,Theorem 1.3,Theorem 6.1]. We explain below the main differences between our present situation and the results proved in [H].

Proof of the main theorem
Proof. For m 1 = m 2 + 1 it is clear as the two g-modules are isomorphic. Let us suppose that m 1 ≥ m 2 + 2. It follows directly from the following statement that we establish. The U q (ĝ)-module is simple and occurs as a composition factor in Rep(U q (ĝ)) of For m 1 = m 2 + 2 this is a direct consequence of the T -system in Theorem 1.3 which gives the decomposition of W (i) m 1 −1,q −2