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arXiv · 2412.20757

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

Hyeonjae Choi·Donghyun Kim·Seung Jin Lee
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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.

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Hyeonjae Choi, Donghyun Kim, Seung Jin Lee. 2025-01-26. Lusztig $q$-weight multiplicities and Kirillov-Reshetikhin crystals. https://arxiv.org/abs/2412.20757

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