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

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.