Combinatorial description of Lusztig $q$-weight multiplicity
Seung Jin Lee



§ ABSTRACT 
We conjecture a precise relationship between Lusztig q-weight multiplicities for type C and Kirillov-Reshetikhin crystals. We also define 𝔤𝔩_n-version of q-weight multiplicity for type C and conjecture the positivity.

Fair Minimum Representation Clustering via Integer Programming A preliminary version of this paper appeared in CPAIOR 2024.
    [
    
===========================================================================================================================





§ INTRODUCTION


The Kostka-Foulkes polynomials K_λμ(t) are coefficients for the modified Hall-Littlewood polynomials when expressed in the Schur basis. They are primarily studied in algebraic combinatorics and representation theory. In 1978, Lascoux and Schützenberger <cit.> showed that 

    K_λμ(t)=∑_T∈SSYT(λ,μ)t^charge(T),
 where the sum is taken over all semistandard Young tableaux of shape λ and weight μ. Since the discovery of the charge formula <cit.>, there have been various interpretations of the charge statistic.

Kostka-Foulkes polynomials can be generalized in mainly two ways. One important generalization is the Macdonald-Kostka polynomials K_λμ(q, t), which are the coefficients for the modified Macdonald polynomials expressed in terms of Schur functions. When q=0, the Macdonald-Kostka polynomial reduces to the Kostka-Foulkes polynomial. Regarding positivity, Haiman <cit.> showed that K_λμ(q, t) lies in ℤ_≥ 0[q, t] by investigating Hilbert schemes. Macdonald polynomials can also be expanded positively in terms of LLT polynomials <cit.>, and Haiman and Grojnowski <cit.> demonstrated that LLT polynomials are Schur positive by proving that the coefficients appear in the theory of Hecke algebras and Kazhdan-Lusztig polynomials, which are known to be nonnegative. Despite various proofs of the positivity of Kostka-Foulkes polynomials, no combinatorial (manifestly positive) formula is currently known.

Another way to generalize the Kostka polynomials is to consider those for other types, known as Lusztig's q-weight multiplicities <cit.>, which are the main focus of this paper. For a simple Lie algebra 𝔤 and dominant weights λ and μ, the Lusztig's q-weight multiplicity ^𝔤_λμ(q) is defined by the formula 
    D_w_0(e^λ∏_α∈Δ^+1/1 - q e^α) = ∑_μ^𝔤_λμ(q) χ^μ,
 where w_0 is the longest element in the Weyl group of 𝔤, Δ^+ is the set of positive roots, D_w is the Demazure operator, and χ^μ is the irreducible character for 𝔤 indexed by μ.  Note that D_w_0(e^λ) = χ^λ is the Weyl character formula. Lusztig's q-weight multiplicities are known to be nonnegative due to the theory of affine Kazhdan-Lusztig polynomials, but manifestly positive formulas are not generally known. For type C, Lecouvey <cit.> constructed the cyclage on Kashiwara-Nakajima tableaux <cit.> and conjectured that the statistics defined by the cyclage might provide a manifestly positive formula for Lusztig's q-weight multiplicity in type C. Additionally, Lecouvey and Lenart <cit.> found a combinatorial formula for ^C_n_λμ(q) when μ = 0.

In this paper, we describe the precise relationship between Lusztig's q-weight multiplicities for type C_n and the energy functions in the affine KR crystals for affine type B^(1)_g for large g. Note that Nakayashiki and Yamada <cit.> demonstrated that the charge for type A corresponds to the energy function in the affine KR crystals for affine type A. To describe the conjecture, we introduce semistandard oscillating tableaux (SSOT) and their realization as certain classical highest weights in the tensor products of Kirillov-Reshetikhin column crystals.

Let 𝔤 be an affine Lie algebra and U'_q(𝔤) the corresponding quantum algebra without derivation. Kirillov-Reshetikhin crystals (KR crystals) are crystal bases B^r,s <cit.> of a certain subset of irreducible finite-dimensional U'_q(𝔤)-modules known as Kirillov-Reshetikhin (KR) modules W^r,s <cit.>. For a partition γ, we define the tensor products of KR column crystals B^t_γ as B^γ_1,1⊗⋯⊗ B^γ_n,1. For each element r in B^t_γ, we can associate a non-negative integer E(r), called the energy, which is invariant under the Kashiwara operators e_i and f_i for i ≠ 0. In our specific case, 𝔤 is of type B^(1)_g for large g, but it will be evident that the idea can be generalized to other types (see Section 3 for the stable version).

On the other hand, the author <cit.> constructed a bijection between King tableaux of shape λ and weight μ and SSOTs of shape λ̃ and μ̃ with at most g columns, where λ̃ is the rectangular complement partition in (g^n) for large g. Note that the number of such objects is equal to the weight multiplicity of weight μ in the irreducible 𝔰𝔭_n-module indexed by λ.

The goal of this paper is to define a certain statistic E(T), also called energy, for each SSOT T by making an injective map from the set of SSOT's of shape λ̃ and μ̃ to the set of some classical highest weights in B^t_μ̃ of weight λ and then take the energy function defined on B^t_μ̃. To be more precise, an SSOT consists of oscillating horizontal strip (α,β,γ) where
α, β, γ are partitions such that α and γ are contained in β, β/α and β/γ are horizontal strips, and the size of a oscillating horizontal strip is ℓ=2|β|-|α|-|γ| where |α| is defined by the sum of parts of α. For each oscillating horizontal strips (α,β,γ), we associate an element in B^ℓ,1 which has an entry i if and only if there is a cell at i-th column of β/α, and has an entry i if and only if there is a cell at i-th column of β/γ. For example, for an oscillating horizontal strip ((1),(3,1),(2,1)) we associate an element 

    1
    2
    3
    3=
        1
    2

 in B^4,1. The right-hand side uses the sage notation, making the column admissible. 

Let ϵ^C(T) be the maximum number of columns among partitions in the SSOT T. In KR crystal language, ϵ^C(T) is the maximum number i appearing in the element T in B^t_μ̃. Then we have the following conjecture:



    ^C_n_λ,μ(q)=∑_T q^E(T)

where the sum runs over all SSOT T of shape λ̃ and weight μ̃ and ϵ^C(T)≤ g.


In general, defining and computing the energy function can be challenging, so we provide an example where the energy function is easy to compute, which is the case μ̃=(1^n).
For an element 

    a_n
    ⊗ 
    a_n-1⊗⋯⊗
    a_1

in (B^1,1)^⊗ n, written with the sage notation, we define the energy function by ∑_i=1^n-1 (n-i)H(a_i+1,a_i) where


    H(b,a)=
        2     if  a=1  and  b=1
    
        1     if  b ≽ a  and  (b,a)≠ (1,1) 
    
        0     if  b ≺ a

under the order

    1 ≺ 2 ≺⋯ n ≺n≺⋯≺1.

Note that when all a_i's are positive, the energy function is the same as the charge statistic of a standard young tableau in Lascoux-Schützenberger formula. In general, Conjecture <ref> generalizes Lascoux-Schützenberger's charge formula for semistandard Young tableaux, which applies to the case where |λ| = |μ|. This conjecture is a significant generalization of the charge formula for type A, as it not only describes the q-weight multiplicities for type C but also establishes a connection with affine crystal theory.
Also note that Conjecture <ref> naturally implies the following monotonicity:

    _λ+(1^n),μ+(1^n)^C_n(q)-_λ,μ^C_n(q) ∈ℤ_≥ 0[q].


The following is an easy example of Conjecture <ref>


    Let n=3, λ=(1,1) and μ=∅. When g=1, λ̃=(1) and μ̃=(1,1,1) and all classcal highest weights in ( B^1,1)^⊗ 3 with the weight λ̃ are

    
    1⊗ 1 ⊗ 1,    1 ⊗1⊗ 1,   2⊗ 2⊗ 1.

    Since the values of ϵ^C for the first two elements are 1 and the value of ϵ^C for the third element is 2, with corresponding energies being 2, 4, and 3, respectively, we have the following:
    
    
    _(1,1),∅^C_n(q)=q^2+q^4.

    Similarly, we have
    
    _(2,2,1),(1,1,1)^C_n(q)=q^2+q^3+q^4.


In Section 3, we prove the conjecture for the stable case by relating q-weight multiplicities to a one-dimensional sum and the X  = ^∞ theorem by Shimozono and Lecouvey <cit.>. The proof in Section 3 applies not only to type C but also to all nonexceptional types.

During the course of this work, the author identified another natural version of the q-weight multiplicity, which we term the 𝔤𝔩_n q-weight multiplicities for nonexceptional types. Consider a function L_A on the set of positive roots, where L_A(α) = 1 if α is a positive root of type A, namely ϵ_i - ϵ_j for i < j, and 0 otherwise. Then the 𝔤𝔩_n q-weight multiplicity for λ,μ of type 𝔤 is defined by the following:


For a function L from the set of positive roots to ℚ, define

    D_w_0(e^λ∏_α∈Δ^+1/1 - q^L(α) e^α) := ∑_μ^𝔤, L_λμ(q) χ^μ.

Then 𝔤𝔩_n q-weight multiplicity for λ,μ of type 𝔤 is defined by _λμ^𝔤,L_A


^C_n,_A_λ,μ(q) is in ℤ_≥ 0[q].



Proofs of both Conjecture <ref> and Conjecture <ref> will be presented in a separate paper <cit.>. In this paper, we investigate the connection between Conjectures <ref> and <ref> and known results, particularly in relation to affine crystal theory and X=^∞ theorem. 



§ ACKNOWLEDGEMENT

The author formulated Conjecture <ref> around December 2019 and shared with Mark Shimozono and Jaehoon Kwon. I thank them for keeping our discussions confidential and for their valuable insights. I also appreciate Hyunjae Choi, Donghyun Kim for helpful discussions. This work is supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No.0450-20240021)



§ LUSZTIG Q-WEIGHT MULITIPLICITY AND 1-DIMENSIONAL SUM

Let 𝔤 be an affine algebra for nonexceptional type. In this section, we explain X=^∞ theorem briefly and relate the theorem with Conjecture <ref> and <ref>.


First, note that Equation (<ref>) is equivalent to the follwing formula:

    ^𝔤,L_λμ(q)= ∑_w∈ W (-1)^w [e^w(λ+ρ)-μ-ρ]∏_α∈Δ^+1/1-q^L(α)e^α

where [e^β]f denotes the coefficient of e^β in f ∈ℤ[P]. The stable version is defined by

    ^∞^𝔤,L_λμ(q)= ∑_w∈ S_n (-1)^w [e^w(λ+ρ)-μ-ρ]∏_α∈Δ^+1/1-q^L(α)e^α,

where S_n is the symmetric group, as a subgroup of W.
The reason why this definition is called stable version is because for large k, we have

    ^∞^𝔤,L_λμ(q)=^𝔤,L_λ+(k^n),μ+(k^n)(q).



One-dimensional (1-d) sums X are graded tensor product multiplicities for affine Kac-Moody algebras, which arise from two-dimensional solvable lattice models <cit.>, and which may be defined using the combinatorics of affine crystal graphs <cit.>. For any nonexceptional family of affine algebras, the 1-d sums have a large rank limit which is called the stable 1-d sums. There are only four distinct kinds of stable 1-d sums <cit.>, and they are labeled by the four partitions ◊∈{∅,(1),(2),(1,1)} having at most 2 cells. See <cit.> for more description. To explain Conjecture <ref> we only need the case ◊=(2) which corresponds to the case when 𝔤 is of type B^(1)_g , A^(2)_2g-1 or D^(1)_g for large g. To be precise, let 

    B_μ=B^1,μ_1⊗⋯⊗ B^1,μ_ℓ


    B^t_μ=B^μ_1,1⊗⋯⊗ B^μ_ℓ,1

where ℓ=ℓ(μ).

For any ◊ and for any tensor product B of KR modules, the X polynomial is defined by

    X^◊_λ,B= ∑ q^E(T)

where T runs over all classical highest elements in B for type ◊, of weight λ. In this paper, B is either B_μ or B^t_μ.

The following is X=^∞ theorem in <cit.>.

 Let ||μ||=∑_i=1^n (i-1)μ_i. Then we have

    X^◊_λ,B_μ(q)=q^||μ||+|μ|-|λ|^∞^𝔤_n_λ̃μ̃(q^-1)







§ PROOF OF CONJECTURE <REF> AND <REF> FOR THE STABLE CASE


There is a follwing duality in X.
<cit.>

    X^◊^t_λ^t,B^t_μ(q)=q^||μ||+|μ|-|λ|X^◊_λ,B_μ(q^-1)


Therefore, we have 

    ^∞^𝔤_n_λμ(q)=X^(1,1)_λ̃,B^t_μ̃(q)

by Theorem <ref> and <ref>, and Conjecture <ref> for the stable case follows.

To prove Conjecture <ref> for the stable case, consider the automorphism ϕ(e^ϵ_i)=q^1/2 e^ϵ_i. Note that the automorphism ϕ commutes with the action of S_n, but not with the action of W. For type C positive roots, ϕ(e^α) is e^α if α is of type A, or q e^α otherwise. Therefore, we have 
    ϕ(e^w(λ))=ϕ(e^λ)= q^⟨λ,ω_n⟩/2 e^λ

where w∈ S_n and ω_n=(1^n). Also, we have

    ϕ(∏_α∈Δ^+1/1-q^L_A(α)e^α)=∏_α∈Δ^+1/1-qe^α.

Therefore, we have
 Assume that 𝔤 is nonexceptional. Then 

    q^|λ|-|μ|^∞_λ,μ^𝔤,L_A(q)=^∞_λ,μ^𝔤(q).


Therefore, Conjecture <ref> for the stable case holds.

There is no such an elegant identity in <ref> for non-stable cases, because ϕ(e^w(α)) is not equal to ϕ(e^α) for w∈ W in general.



§ FURTHER DIRECTIONS

One natural question is whether Conjectures <ref> and <ref> can be generalized to other types. Partial progress on this topic will appear in <cit.>. Applications of Conjecture <ref> include its connections with parabolic q-weight multiplicities <cit.>, type C analogues <cit.> of Catalan functions <cit.>, and potentially with Macdonald polynomials and LLT polynomials for type C.

Another question is whether there is a geometric or representation-theoretic interpretation of the duality that appears in Conjecture <ref>. It would be interesting to explain Theorem <ref> without relying on the X = K result, where K denotes Kostka polynomials. The proof of Theorem <ref> utilizes the X = K theorem and the duality for Kostka polynomials.

Lastly, Conjecture <ref> is not compatible with Lecouvey's conjecture in <cit.>, but it seems to be compatible with the work <cit.> by Lecouvey and Lenart when μ=0. It would be interesting to generalize the computation of the energy in their way.

alpha  
 


