A skein-like multiplication algorithm for unipotent Hecke algebras

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