Lusztig q-weight multiplicities extend the Kostka-Foulkes polynomials to a broader range of Lie types. In this talk, we investigate these multiplicities through the framework of Kirillov--Reshetikhin crystals. For type C with dominant weights and type B with dominant spin weights, we give combinatorial formulas for Lusztig's q-weight multiplicities in terms of energy functions on Kirillov--Reshetikhin crystals. These formulas generalize the charge formula for Kostka-Foulkes polynomials in type A.

We also introduce level-restricted q-weight multiplicities for nonexceptional types and prove their positivity by giving combinatorial formulas. We conclude with two further generalizations, namely a type C Catalan-Kostka generalization using root ideals and a noncommutative generalization using noncommutative Kostka polynomials and immaculate tableaux. 

발표시간 : 13:00 ~ 14:00