The structure of normal lattice supercharacter theories

TL;DR

This paper explores a supercharacter theory for finite groups that simplifies character analysis by translating core questions into combinatorial lattice problems, balancing detail retention with computational tractability.

Contribution

It introduces a specific supercharacter theory framework that bridges detailed character theory and combinatorial lattice structures, offering a new approach to group analysis.

Findings

Provides a lattice-based framework for supercharacter theories

Simplifies the enumeration and analysis of characters

Balances information retention with computational simplicity

Abstract

The character theory of finite groups has numerous basic questions that are often already quite involved: enumerating of irreducible characters, their character formulas, point-wise product decompositions, and restriction/induction between groups. A supercharacter theory is a framework for simplifying the character theory of a finite group, while ideally not losing all important information. This paper studies one such theory that straddles the gap between retaining valuable group information while reducing the above fundamental questions to more combinatorial lattice constructions.