Hyeonjae Choi, Donghyun Kim, Seung Jin Lee
Published 2024-12-30Version 1
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.
Comments: 60 pages, comments are welcome
Inversions of Semistandard Young Tableaux
A Combinatorial Formula for the Character of the Diagonal Coinvariants
Perfectness of Kirillov-Reshetikhin Crystals $B^{r,s}$ for types $E_{6}^{(1)}$ and $E_{7}^{(1)}$ with a minuscule node $r$
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