Determining Aschbacher classes using characters

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.