A skein-like multiplication algorithm for unipotent Hecke algebras

Abstract

Let G G be a finite group of Lie type (e.g. G L n ( F q ) GL_n(\mathbb {F}_q) ) and U U a maximal unipotent subgroup of G G . If ψ \psi is a linear character of U U , then the unipotent Hecke algebra is H ψ = E n d C G ( I n d U G ( ψ ) ) \mathcal {H}_\psi =\mathrm {End}_{\mathbb {C}G} (\mathrm {Ind}_U^G(\psi )) . Unipotent Hecke algebras have a natural basis coming from double cosets of U U in G G . This paper describes relations for reducing products of basis elements, and gives a detailed description of the implications in the case G = G L n ( F q ) G=GL_n(\mathbb {F}_q) .