Nonstandard braid relations and Chebyshev polynomials

Abstract

A fundamental open problem in algebraic combinatorics is to find a positive combinatorial formula for Kronecker coefficients, which are multiplicities of the decomposition of the tensor product of two \S_r-irreducibles into irreducibles. Mulmuley and Sohoni attempt to solve this problem using canonical basis theory, by first constructing a nonstandard Hecke algebra B_r, which, though not a Hopf algebra, is a u-analogue of the Hopf algebra \CC \S_r in some sense (where u is the Hecke algebra parameter). For r=3, we study this Hopf-like structure in detail. We define a nonstandard Hecke algebra \bar{\H}^{(k)}_3 \subseteq \H_3^{\tsr k}, determine its irreducible representations over \QQ(u), and show that it has a presentation with a nonstandard braid relation that involves Chebyshev polynomials evaluated at \frac{1}{u + u^{-1}}. We generalize this to Hecke algebras of dihedral groups. We go on to show that these nonstandard Hecke algebras have bases similar to the Kazhdan-Lusztig basis of \H_3 and are cellular algebras in the sense of Graham and Lehrer.