Determining Aschbacher classes using characters

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.