Presenting Hecke endomorphism algebras by Hasse quivers with relations

TL;DR

This paper provides a quiver with relations presentation for Hecke endomorphism algebras associated with arbitrary Coxeter groups, extending previous work and offering algorithms for their explicit construction.

Contribution

It introduces a new presentation method for Hecke endomorphism algebras using Hasse quivers with relations, applicable to all Coxeter groups.

Findings

Presented a quiver with relations for Hecke endomorphism algebras

Developed an algorithm to find torsion relations over z[q]

Determined torsion relations for rank 2 groups and S_4

Abstract

A Hecke endomorphism algebra is a natural generalisation of the $q$-Schur algebra associated with the symmetric group to a Coxeter group. For Weyl groups, B. Parshall, L. Scott and the first author \cite{DPS,DPS4} investigated the stratification structure of these algebras in order to seek applications to representations of finite groups of Lie type. In this paper we investigate the presentation problem for Hecke endomorphism algebras associated with arbitrary Coxeter groups. Our approach is to present such algebras by quivers with relations. If $R$ is the localisation of $\mathbb Z[q]$ at the polynomials with the constant term 1, the algebra can simply be defined by the so-called idempotent, sandwich and extended braid relations. As applications of this result, we first obtain a presentation of the 0-Hecke endomorphism algebra over $\mathbb{Z}$ and then develop an algorithm for presenting the Hecke endomorphism algebras over $\mathbb Z[q]$ by finding torsion relations. As examples, we determine the torsion relations required for all rank 2 groups and the symmetric group $\mathfrak{S}_4$.