On the centre of Iwahori-Hecke algebras

TL;DR

This paper proves that the center of certain Hecke algebras is trivial, based on a structural analysis of conjugacy classes in infinite Coxeter groups, revealing new insights into their algebraic properties.

Contribution

It establishes the triviality of the center for Hecke algebras of non-finite, non-affine types, using a novel conjugacy class structure result in Coxeter groups.

Findings

Center of these Hecke algebras is trivial.

Conjugacy classes in infinite Coxeter groups exhibit unbounded length growth.

Structural properties of Coxeter groups influence algebraic centers.

Abstract

We prove triviality of the centre of arbitrary Hecke algebras of irreducible non-finite non-affine type. This result is obtained as a consequence of the following structure result for conjugacy classes of the underlying Coxeter groups. If $W$ is any infinite irreducible Coxeter group and $w \in W$ is a nontrivial element that is assumed not be a translation in case $W$ is affine, then there is an infinite sequence of conjugates of $w$ by Coxeter generators whose length is non-decreasing and tends to infinity.