Supercharacters of adjoint groups of radical rings and related subgroups

Abstract: Describing the conjugacy classes and/or irreducible characters of the unitriangular group over a finite field is known to be an impossibly difficult problem. Superclasses and supercharacters have been introduced (under the names of "basic varieties" and "basic characters") as an attempt to approximate conjugacy classes and irreducible characters using a cruder version of Kirillov's method of coadjoint orbits.

In the past thirty years, these notions have been recognised in several areas (seemingly unrelated to representation theory): exponential sums in number theory, random walks in probability and statistics, association schemes in algebraic combinatorics...

In this talk, we will describe and illustrate the main ideas and recent developments of the standard supercharacter theory of adjoint groups of radical rings. We will explore the close relation to Schur rings, and extend a well-known factorisation of supercharacters of unitriangular groups which explains the alternative definition as basic characters.

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar