Critical groups of group representations

TL;DR

This paper studies the critical group associated with faithful representations of finite groups, providing formulas for its order, restrictions on its structure, and explicit computations for specific cases like regular and reflection representations.

Contribution

It introduces new formulas for the order of the critical group and determines its structure for key classes of representations, advancing understanding of their algebraic properties.

Findings

Order of the critical group expressed via character values

Restrictions on the subgroup structure of the critical group

Explicit structure computed for regular and reflection representations

Abstract

This paper investigates the critical group of a faithful representation of a finite group. It computes the order of the critical group in terms of the character values, and gives some restrictions on its subgroup structure. It also computes the precise structure of the critical group both for the regular representation of any finite group, and for the reflection representation of the symmetric group.