The University of Michigan Combinatorics Seminar


Abstract 

The Kronecker coefficient g_{&lambda&mu&nu} is the multiplicity of an irreducible S_{r}module M_{&nu} in the tensor product M_{&lambda} &otimes M_{&mu}. A difficult open problem in algebraic combinatorics is to find a positive combinatorial formula for these coefficients. I will describe Mulmuley and Sohoni's approach to this problem using the nonstandard Hecke algebra and quantum group. The nonstandard Hecke algebra is a subalgebra of the tensor square of the Hecke algebra, and the nonstandard quantum group is defined through its coordinate ring, which is a quotient of the free algebra C(q)< u_{ij}> by certain quadratic relations. I will discuss the representation theory of these algebras and how they might help solve the Kronecker problem. Specifically, I will describe the irreducible representations of the nonstandard Hecke algebra in the tworow case and give evidence that these have nice bases which give rise to filtrations into S_{r}irreducibles at q=1. 