Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Galois Automorphisms and Classical Groups

A. A. Schaeffer Fry, Jay Taylor

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.

Galois Automorphisms and Classical 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.

Paper Structure

This paper contains 10 sections, 39 theorems, 34 equations, 1 table.

Table of Contents

  1. Introduction
  2. Automorphisms
  3. Galois Automorphisms
  4. Galois Automorphisms Relevant for the McKay--Navarro Conjecture
  5. Power Maps and Regular Unipotent Elements
  6. A Variation of GGGCs for Disconnected Groups
  7. Classical Groups
  8. Action of Automorphisms on Unipotent Characters
  9. Power Maps and Classical Groups
  10. Cuspidal Characters of Orthogonal Groups

Key Result

Theorem A

Assume $\mathbf{G} = \mathop{\mathrm{SO}}\nolimits(V)$ is a special orthogonal group defined over $\mathbb{F}_q$ with $q$ odd. Let $F_p,\gamma : \mathbf{G} \to \mathbf{G}$ be field and graph automorphisms, respectively, with $\gamma$ the identity when $\dim(V)$ is odd. Assume $s \in \mathbf{G}^{\sta

Theorems & Definitions (74)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Lemma 11
  • Proof
  • Remark 12
  • Remark 14
  • Proposition 15
  • Proof
  • ...and 64 more