Irreducible adjoint representations in prime power dimensions, with an application

TL;DR

This paper constructs an infinite family of finite group representations with irreducible adjoint actions and applies this to study lacunarity in Frobenius traces of Galois representations.

Contribution

It introduces a new infinite family of representations with irreducible adjoint actions and applies it to problems in Galois representation theory.

Findings

Constructed an infinite family of such representations.

Provided an application to lacunarity of Frobenius traces.

Enhanced understanding of the structure of Galois representations.

Abstract

We construct an infinite family of representations of finite groups with an irreducible adjoint action and we give an application to the question of lacunary of Frobenius traces in Galois representations.