Level-restricted generalized Kostka polynomials

Generalized Kostka polynomials are $q$-analogues of tensor product multiplicities. They can be expressed as generating functions of highest weight vectors of $U_q(sl_n)$-crystals. Extending the $U_q(sl_n)$-crystal structure to a $U_q(\widehat{sl}_n')$-crystal structure gives rise to level-restricted generalized Kostka polynomials.

We present new general fermionic formulas for the level-restricted generalized Kostka polynomials. Fermionic expressions originate from the Bethe Ansatz and can be combinatorialized by rewriting the expressions as weighted sums over sets of rigged configurations. We sketch the proof of the fermionic formula for the level-restricted generalized Kostka polynomials. To this end we review a recently established statistic-preserving bijection between Littlewood--Richardson tableaux and rigged configurations, and show that it is well-behaved with respect to level-restriction. By taking appropriate limits, the fermionic formulas for the level-restricted generalized Kostka polynomials also give rise to new fermionic formulas for type $A$ branching functions.

This talk is based on the following two preprints: (1) A.N. Kirillov, A. Schilling and M. Shimozono, A bijection between Littlewood-Richardson tableaux and rigged configurations, math.CO/9901037. (2) A. Schilling and M. Shimozono, Fermionic formulas for level-restricted generalized Kostka polynomials and coset branching functions, in preparation.