Supercharacters of discrete algebra groups

TL;DR

This paper extends the supercharacter theory from finite algebra groups to countable discrete algebra groups using ergodic theory, providing a broader framework and illustrating it with upper unitriangular matrices.

Contribution

It introduces a new construction of supercharacters for countable discrete algebra groups, generalizing previous finite group methods with an ergodic theoretical approach.

Findings

Extended supercharacter theory to countable discrete algebra groups.

Characterized supercharacters of upper unitriangular matrices.

Connected supercharacter theory with ergodic theory principles.

Abstract

The concept of a supercharacter theory of a finite group was introduced by Diaconis and Isaacs as an alternative to the usual irreducible character theory, and exemplified with a particular construction in the case of finite algebra groups. We extend this construction to arbitrary countable discrete algebra groups, where superclasses and indecomposable supercharacters play the role of conjugacy classes and indecomposable characters, respectively. Our construction can be understood as a cruder version of Kirillov's orbit method and a generalisation of Diaconis and Isaacs construction for finite algebra groups. However, we adopt an ergodic theoretical point of view. The theory is then illustrated with the characterisation of the standard supercharacters of the group of upper unitriangular matrices over an algebraic closed field of prime characteristic.