Lusztig $q$-weight multiplicities and Kirillov-Reshetikhin crystals

TL;DR

This paper develops combinatorial formulas for Lusztig $q$-weight multiplicities using Kirillov-Reshetikhin crystals, extending known statistics to broader Lie types and establishing positivity results.

Contribution

It introduces a new combinatorial framework for Lusztig $q$-weight multiplicities across multiple Lie types, generalizing existing charge statistics.

Findings

Derived combinatorial formulas for type C and B multiplicities.

Established positivity of level-restricted $q$-weight multiplicities.

Extended charge statistic to non-type A Lie algebras.

Abstract

Lusztig $q$-weight multiplicities extend the Kostka-Foulkes polynomials to a broader range of Lie types. In this work, we investigate these multiplicities through the framework of Kirillov-Reshetikhin crystals. Specifically, for type $C$ with dominant weights and type $B$ with dominant spin weights, we present a combinatorial formula for Lusztig $q$-weight multiplicities in terms of energy functions of Kirillov-Reshetikhin crystals, generalizing the charge statistic on semistandard Young tableaux for type $A$. Additionally, we introduce level-restricted $q$-weight multiplicities for nonexceptional types, and prove positivity by providing their combinatorial formulas.