Levelrestricted generalized Kostka polynomialsAnne SchillingMIT
September 24,

ABSTRACT


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 levelrestricted generalized Kostka polynomials. We present new general fermionic formulas for the levelrestricted 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 levelrestricted generalized Kostka polynomials. To this end we review a recently established statisticpreserving bijection between LittlewoodRichardson tableaux and rigged configurations, and show that it is wellbehaved with respect to levelrestriction. By taking appropriate limits, the fermionic formulas for the levelrestricted 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 LittlewoodRichardson tableaux and rigged configurations, math.CO/9901037. (2) A. Schilling and M. Shimozono, Fermionic formulas for levelrestricted generalized Kostka polynomials and coset branching functions, in preparation. 
Combinatorics Seminar, Mathematics Department, MIT, sara(atsign)math.mit.edu 

