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.

v1. 40 pages; v2. 42 pages. Corrected the statement of Thm. C and updated Section 16 to reflect this change. No other changes; v3. Final version

References in corpus (2)

• Unitriangular Shape of Decomposition Matrices of Unipotent Blocks
• Galois action on the principal block and cyclic Sylow subgroups

Cited by in corpus (2)

• On Héthelyi-Külshammer's conjecture for principal blocks
• Galois-equivariant McKay bijections for primes dividing $q-1$