Character Theory for Semilinear Representations

TL;DR

This paper develops a character theory for semilinear representations of groups acting on fields, generalizing classical character theory to a broader algebraic context.

Contribution

It introduces a framework to describe semilinear representations via linear ones, extending character theory to new algebraic settings.

Findings

Category of semilinear representations described in terms of linear representations of a kernel group

Provides a character theory for semilinear representations when the group is finite and the field has characteristic zero

Recovers classical character theory as a special case when the group action on the field is trivial

Abstract

Let $G$ be a group acting on a field $L$, and suppose that $L /L^G$ is a finite extension. We show that the category of semilinear representations of $G$ over $L$ can be described in terms of the category of linear representations of $H$, the kernel of the map $G \rightarrow \mathrm{Aut}(L)$. When $G$ is finite and $L$ has characteristic 0 this provides a character theory for semilinear representations of $G$ over $L$, which recovers ordinary character theory when the action of $G$ on $L$ is trivial.