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Presenting Hecke endomorphism algebras by Hasse quivers with relations

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 $\mathfrak{S}_4$.