Nonstandard Braid Relations and Chebyshev Polynomials

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 H̄ (k) 3 ⊆ H ⊗k 3 , 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 1 u+u . 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.