A reduction theorem for the Galois–McKay conjecture

Abstract

We introduce H-triples and a partial order relation on them, generalizing the theory of ordering character triples developed by Navarro and Späth. This generalization takes into account the action of Galois automorphisms on characters and, together with previous results of Ladisch and Turull, allows us to reduce the Galois-McKay conjecture to a question about simple groups.2010 Mathematics Subject Classification. Primary 20C15; Secondary 20C25.

Galois-equivariant McKay bijections for primes dividing q − 1

“…In particular, this would imply that the corresponding fields of values are preserved over the field of ℓ-adic numbers (see also [Tur08]). This version of the conjecture was recently reduced by Navarro-Späth-Vallejo in [NSV20] to proving certain inductive conditions for finite simple groups. These "inductive Galois-McKay conditions" can roughly be described as an "equivariant" condition and an "extension" condition.…”

The inductive McKay--Navarro condition for the Suzuki and Ree groups

“…For the McKay-Navarro conjecture, a reduction theorem has been proven in 2019 by Navarro, Späth, and Vallejo [NSV20]. The resulting inductive condition has been verified for all finite groups of Lie type in their defining characteristic [Ruh21] [Joh20] and for p = 2 and the simple groups C n (q) (n ≥ 1), B n (q) (n ≥ 3, (n, q) = (3, 3)), G 2 (q) ( 3 ∤ q), 3 D 4 (q), F 4 (q), E 7 (q), and E 8 (q) where q is a power of an odd prime [RSF21].…”

The inductive McKay--Navarro conditions for the prime 2 and some groups of Lie type

“…Here we expand on the work of [SF21] to complete the proof that when ℓ = 2, the inductive McKay-Navarro (also called inductive Galois-McKay) conditions from [NSV20] are satisfied for the untwisted groups of Lie type that do not admit graph automorphisms, as well as the group 3 D 4 (q). Theorem A.…”