Towards a Combinatorial Model for $q$-weight Multiplicities of Simple Lie Algebras (Extended Abstract)

TL;DR

This paper introduces a new combinatorial model for q-weight multiplicities in simple Lie algebras, using Kostant partitions and sign-reversing involutions to achieve positive expansions, aiming to extend beyond type A.

Contribution

It proposes a novel approach based on Kostant partitions and involutions to model Kostka-Foulkes polynomials, potentially applicable to all classical Lie types.

Findings

Established a positive expansion using Kostant partitions

Defined a simple statistic as the number of parts in partitions

Provided a new combinatorial framework for q-weight multiplicities

Abstract

Kostka-Foulkes polynomials are Lusztig's $q$-analogues of weight multiplicities for irreducible representations of semisimple Lie algebras. It has long been known that these polynomials have non-negative coefficients. A statistic on semistandard Young tableaux with partition content, called \textit{charge}, was used to give a combinatorial formula exhibiting this fact in type $A$. Defining a charge statistic beyond type $A$ has been a long-standing problem. Here, we take a completely new approach based on the definition of Kostka-Foulkes polynomials as an alternating sum over Kostant partitions, which can be thought of as formal sums of positive roots. We use a sign-reversing involution to obtain a positive expansion, in which the relevant statistic is simply the number of parts in the Kostant partitions. The hope is that the simplicity of this new crystal-like model will naturally extend to other classical types.