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 Sr-irreducibles into irreducibles. Mulmuley and Sohoni attempt to solve this problem using canonical basis theory, by first constructing a nonstandard Hecke algebra Br, which, though not a Hopf algebra, is a u-analogue of the Hopf algebra CSr 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 Hˇ3(k)⊆H3⊗k, determine its irreducible representations over Q(u), and show that it has a presentation with a nonstandard braid relation that involves Chebyshev polynomials evaluated at 1u+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 H3 and are cellular algebras in the sense of Graham and Lehrer.