The University of Michigan Combinatorics Seminar
Winter 2003
February 14, 4:10-5:00, 3866 East Hall

More fermionic formulae

Michael Zabrocki

York University


For g, a Kac-Moody Lie algebra of nonexceptional affine type, Hatayama, Kuniba, Okado, Takagi, Tsuboi and Yamada, conjectured that a certain family of finite-dimensional U'q(g)-modules are tensor products of modules and their graded multiplicities are called one-dimensional sums X. The above authors also define the fermionic formula M in terms of combinatorial objects called rigged configurations and conjecture that X=M.

As the rank of g goes to infinity (with g staying within the same affine family), the graded characters of tensor products of these modules have stable limits which can be identified with symmetric functions. We call these symmetric functions the universal affine characters. Of the seven infinite families of affine root systems, there are only four different families of universal affine characters. In type A(1)n they are generalizations of the Hall-Littlewood symmetric functions and the coefficients are known as generalized (parabolic) Kostka polynomials.

We define three deformations of the generalized Kostka polynomials and conjecture that these deformed polynomials are the other three families of universal affine characters. Let us call the coefficients appearing in these symmetric function expressions K and we conjecture that X=M=K.

This is joint work with Mark Shimozono.