Representations of Hecke algebras on quotients of path algebras

TL;DR

This paper constructs universal representations of multi-parameter Hecke algebras on quotients of path algebras, exploring their connections with W-graph representations and ideals related to Lusztig's asymptotic Hecke algebra.

Contribution

It introduces a new framework for representing Hecke algebras via quotients of path algebras and relates these to W-graph representations and Lusztig's asymptotic algebra.

Findings

Established universal representations on quotient path algebras.

Connected quotient path algebras to Lusztig's asymptotic Hecke algebra.

Provided a method to generate ideals for quotient constructions.

Abstract

Let $(W,S)$ be a Coxeter system. A $W$-graph encodes a representation of the Hecke algebra $\mathcal{H}$ of $W$. We construct universal representations of multi-parameter Hecke algebras on certain quotients of path algebras, and study their relationships with $W$-graph representations. We also study the quotients of path algebras on their own, motivated by one example where the quotient path algebra is isomorphic to an ideal of Lusztig asymptotic Hecke algebra. Finally, we describe a method to obtain a generating set for the ideals by which we quotient the path algebras.