The University of Michigan Combinatorics Seminar
Fall 2010
October 15, 4:10-5:00, 3866 East Hall



Nonstandard Hecke algebra for the Kronecker problem

Jonah Blasiak

University of Michigan


Abstract

The Kronecker coefficient g&lambda&mu&nu is the multiplicity of an irreducible Sr-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 two-row case and give evidence that these have nice bases which give rise to filtrations into Sr-irreducibles at q=1.