Galois Automorphisms and Classical Groups

TL;DR

This paper extends the understanding of automorphism actions on irreducible characters of finite symplectic and special orthogonal groups, providing explicit character field descriptions through Galois automorphisms.

Contribution

It generalizes previous work by describing Galois automorphism actions on characters and extends results to special orthogonal groups.

Findings

Complete description of Galois automorphism action on characters

Explicit character field descriptions for symplectic groups

Extension of character analysis to special orthogonal groups

Abstract

In a previous work, the second-named author gave a complete description of the action of automorphisms on the ordinary irreducible characters of the finite symplectic groups. We generalise this in two directions. Firstly, using work of the first-named author, we give a complete description of the action of Galois automorphisms on irreducible characters. Secondly, we extend both descriptions to cover the case of special orthogonal groups. As a consequence, we obtain explicit descriptions for the character fields of symplectic and special orthogonal groups.