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

Abstract

We verify the inductive McKay-Navarro condition for the Suzuki and Ree groups for all primes as well as for p = 3 and the groups PSL 3 (q) with q ≡ 4, 7 mod 9, PSU 3 (q) with q ≡ 2, 5 mod 9, and G 2 (q) with q = 2, 4, 5, 7 mod 9. On the way, we show that there exists a Galois-equivariant Jordan decomposition for the irreducible characters of the Suzuki and Ree groups.2010 Mathematics Subject Classification. 20C15, 20C33.

On the inductive McKay–Navarro condition for finite groups of Lie type in their defining characteristic

“…Proof. For ℓ = 2 and ℓ = 3 we already know that the statement holds by [Joh22b] and [Joh21]. The group 3.G 2 (3) can be considered with [GAP19] and we see that Condition 1 holds.…”

“…We consider the inductive McKay-Navarro condition for a prime ℓ dividing |G| = q 6 (q − 1) 2 (q + 1) 2 (q 2 + q + 1)(q 2 − q + 1) = q 6 φ 2 1 φ 2 2 φ 3 φ 6 where φ d denotes the d-th cyclotomic polynomial evaluated at q. We assume that ℓ ≥ 5 since ℓ = 2 and ℓ = 3 have already been considered in [Joh21] and [Joh22b].…”

“…In [NSV20], Navarro, Späth, and Vallejo proved a reduction theorem for the McKay-Navarro conjecture, yielding an inductive condition that should be verified for all finite simple groups. So far, this has been done for groups of Lie type in their defining characteristic in [Ruh21] and [Joh22b] as well as for the Suzuki and Ree groups and all primes in [Joh21]. Ruhstorfer and Schaeffer Fry further showed in [RSF22b] that the inductive condition holds for all finite simple groups and the prime ℓ = 2 and thereby proved that the conjecture itself is true for ℓ = 2.…”