Supercharacters of unipotent and solvable groups

TL;DR

This paper reviews supercharacter theory, explores its construction for algebra and triangular type groups, and characterizes the Hopf algebra structure of supercharacters for finite triangular groups.

Contribution

It extends supercharacter theory to finite groups of triangular type and characterizes the Hopf algebra structure of their supercharacters.

Findings

Supercharacter theory applied to algebra groups and unitriangular groups.

Construction of supercharacter theory for finite groups of triangular type.

Characterization of Hopf algebra structure of supercharacters.

Abstract

The notion of the supercharacter theory was introduced by P.Diaconis and I.M.Isaaks in 2008. In this paper we review the main statements of the general theory, we observe the construction of supercharacter theory for algebra groups and the theory of basic characters for the unitriangular groups over the finite field. Basing on the previous papers of the author, we construct the supercharacter theory for the finite groups of triangular type. We characterize the structure of Hopf algebra of supercharacters for the triangular group over the finite field.