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

Abstract

For a prime ℓ, the McKay conjecture suggests a bijection between the set of irreducible characters of a finite group with ℓ ′ -degree and the corresponding set for the normalizer of a Sylow ℓsubgroup. Navarro's refinement suggests that the values of the characters on either side of this bijection should also be related, proposing that the bijection commutes with certain Galois automorphisms. Recently, Navarro-Späth-Vallejo have reduced the McKay-Navarro conjecture to certain "inductive" conditions on finite simple groups. We prove that these inductive McKay-Navarro (also called the inductive Galois-McKay) conditions hold for the prime ℓ = 2 for several groups of Lie type, including the untwisted groups without nontrivial graph automorphisms.

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]. Additionally, there are some partial results for groups not of type A in [SF20].…”