Supercharacter theories constructed by the method of little groups

TL;DR

This paper modifies the method of little groups to construct supercharacter theories for semidirect products with abelian normal subgroups, enabling the reproduction of known theories and the creation of new ones for unipotent groups.

Contribution

It introduces a modified method of little groups for supercharacter theory construction, expanding the toolkit for analyzing unipotent groups.

Findings

Reproduces known supercharacter theories of unipotent groups

Constructs new supercharacter theories for upper-triangular matrices

Provides a systematic approach for supercharacter theory construction

Abstract

The method of little groups describes the irreducible characters of semidirect products with abelian normal subgroups in terms of the irreducible characters of the factor groups. We modify this method to construct supercharacter theories of semidirect products with abelian normal subgroups. In particular, we apply this construction to reproduce known supercharacter theories of several families of unipotent groups. We also utilize our method to construct a collection of new supercharacter theories of the unipotent upper-triangular matrices.