Introduction to Supercharacter Theory

A supercharacter theory for a finite group is a coarsening of ordinary character theory. The theorty was first developed by Carlos Andre (mid '90s) to study the representation theory of the group of upper uni-triangular matrices over a finite field U_n(F_p). (This is a famously "wild" problem, i.e., unapproachable by ordinary character theory.) This talk will serve as a gentle introduction to the theory of supercharacters and superclasses. (We will follow the 2008 paper of Diaconis & Isaacs.) Towards the end of the hour, we mention exciting new work connecting the ring of supercharacters of U_n(F₂ to the Hopf algebra NCSYM of symmetric functions in noncommuting variables. (We will follow an arXiv preprint with 28 authors!!!