Nonstandard braid relations and Chebyshev polynomials

TL;DR

This paper introduces nonstandard braid relations involving Chebyshev polynomials within certain Hecke algebras, providing new algebraic structures and bases that could impact the understanding of Kronecker coefficients in algebraic combinatorics.

Contribution

It constructs and analyzes nonstandard Hecke algebras with braid relations involving Chebyshev polynomials, extending the algebraic framework for combinatorial representation theory.

Findings

Defined nonstandard Hecke algebra ^{(k)}_3 and its irreducible representations

Established nonstandard braid relations involving Chebyshev polynomials

Showed these algebras are cellular with bases similar to Kazhdan-Lusztig bases

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.