Unisingular representations in arithmetic and Lie theory

TL;DR

This paper classifies fixed-point subgroups of symplectic groups over finite fields, focusing on irreducible cases and introducing new infinite series of such subgroups from Lie type representations.

Contribution

It provides a complete classification of irreducible fixed-point subgroups of Sp_8(2) and constructs new infinite series of such subgroups in symplectic groups from Lie theory.

Findings

Classified all irreducible fixed-point subgroups of Sp_8(2).

Constructed new infinite series of fixed-point subgroups in symplectic groups.

Connected fixed-point subgroup properties with representations of Lie type groups.

Abstract

Let G be a subgroup of GL(V), where V is a finite dimensional vector space over a finite field of characteristic p >0. If det(g-1) = 0 for all g \in G then we call G a fixed-point subgroup of GL(V). Motivated in parallel by questions in arithmetic and linear group theory, we classify all irreducible fixed-point subgroups of Sp_8(2) and give new infinite series of irreducible fixed-point subgroups of symplectic groups Sp_m(2) for various m arising from certain representations of groups of Lie type.