Determining Aschbacher classes using characters

2014/02/26 by Sebastian Jambor, Jambor, Sebastian
Mathematics · #20C99 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #math.GR #msc:20C99

paper · pdf · doi:10.48550/arxiv.1402.6395

8 pages

arxiv created 2014/02/26 · openalex publication_date 2014/02/26 · arxiv updated 2014/02/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

Related

• Character Theory for Semilinear Representations
• Harmonic Analysis on GL(n) over Finite Fields
• Representations of the general linear groups which are irreducible over subgroups
• Representations and characters of finite groups
• Simple characters and coefficient systems on the building