# Mathematics Colloquia and Seminars

This talk examines the structure and representation theory of unipotent Hecke algebras. We construct them with the following ingredients: a finite group of Lie type $G$ (such as the general linear group over a finite field), a maximal unipotent subgroup $U$ of $G$ (such as the subgroup of upper-triangular matrices with ones on the diagonal), and a one-dimensional $U$-module $\psi$. The double cosets $U\backslash G/U$ determine a natural basis for unipotent Hecke algebras, whose multiplication relations have a skein-like algorithm. In the case $G=GL_n(F_q)$, the combinatorics of partitions and weighted column strict tableaux governs the representation theory, giving rise to an explicit combinatorial map from the natural basis to pairs of multi-tableaux (also known as an RSK-insertion).