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

Presenting Hecke endomorphism algebras by Hasse quivers with relations