Determining Aschbacher classes using characters

TL;DR

This paper develops criteria based solely on character theory to determine properties of irreducible group representations, including their Aschbacher class membership over finite fields.

Contribution

It introduces new methods to infer representation properties from characters, enabling classification into Aschbacher classes without explicit module analysis.

Findings

Criteria to determine if a representation preserves a form

Conditions for realizability over subfields

Algorithms to identify Aschbacher classes over finite fields

Abstract

Let $\Delta\colon G \to \mathrm{GL}(n, K)$ be an absolutely irreducible representation of an arbitrary group $G$ over an arbitrary field $K$; let $\chi\colon G \to K\colon g \mapsto \mathrm{tr}(\Delta(g))$ be its character. In this paper, we assume knowledge of $\chi$ only, and study which properties of $\Delta$ can be inferred. We prove criteria to decide whether $\Delta$ preserves a form, is realizable over a subfield, or acts imprimitively on $K^{n \times 1}$. If $K$ is finite, this allows us to decide whether the image of $\Delta$ belongs to certain Aschbacher classes.