Brauer Theory for Profinite Groups

TL;DR

This paper extends Brauer theory from finite groups to profinite groups, developing new tools for understanding their representations, characters, and block structures in a more general setting.

Contribution

It generalizes Brauer theory to profinite groups, introducing Grothendieck groups, decomposition maps, and block theory in this broader context.

Findings

Generalization of Brauer theory to profinite groups

Development of functorial Grothendieck groups for profinite groups

A method to compute Cartan matrices using quotient groups

Abstract

Brauer Theory for a finite group can be viewed as a method for comparing the representations of the group in characteristic 0 with those in prime characteristic. Here we generalize much of the machinery of Brauer theory to the setting of profinite groups. By regarding Grothendieck groups as functors we describe corresponding Grothendieck groups for profinite groups, and generalize the decomposition map, regarded as a natural transformation. We discuss characters and Brauer characters for profinite groups. We give a functorial description of the block theory of a profinite group. We finish with a method for computing the Cartan matrix of a finite group $G$ given the Cartan matrix for a quotient of $G$ by a normal $p$-subgroup.