Hecke algebras and involutions in Coxeter groups

TL;DR

This paper explores the relationship between Hecke algebras and involutions in Coxeter groups, providing an alternative algebraic action definition using recent advances in Soergel bimodules.

Contribution

It introduces a new approach to defining the Hecke algebra action on involution-based modules via Elias and Williamson's results on Soergel bimodules.

Findings

New algebraic action of Hecke algebra on involution modules

Connection established between Coxeter group involutions and Soergel bimodules

Alternative framework for studying Hecke algebra representations

Abstract

Let $W$ be a Coxeter group and let $M$ be the free $Z[v,v^{-1}]$-module with basis indexed by the involutions of $W$. We show how recent results of Elias and Williamson on Soergel bimodules can be used to give an alternative definition of an action of the Hecke algebra of $W$ on $M$.