Nonstandard Hecke algebra for the Kronecker problem

The Kronecker coefficient gives the multiplicity of an irreducible S_r-module in the tensor product of two other irreducible modules. 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 associative C(q)-algebra on variables 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 S_r-irreducibles at q=1.