The Isaacs–Navarro conjecture for covering groups of the symmetric and alternating groups in odd characteristic

Abstract

In this paper, we prove that a refinement of the Alperin-McKay Conjecture for p-blocks of finite groups, formulated by I.M. Isaacs and G. Navarro in 2002, holds for all covering groups of the symmetric and alternating groups, whenever p is an odd prime.

Local-global conjectures in the representation theory of finite groups

“…(Note that this is a true refinement of Conjecture 2.2 whenever p ≥ 5.) This has been shown to hold for example for S n , A n and their double covers by Fong [37], Nath [79] and Gramain [42] respectively. Two further refinements on the properties of the required bijection concerning the action of those Galois automorphisms fixing a prime ideal above p were put forward in the same paper [48], and by Navarro [73] respectively.…”

“…Nath [79] and Gramain [42] respectively. Two further refinements on the properties of the required bijection concerning the action of those Galois automorphisms fixing a prime ideal above p were put forward in the same paper [48], and by Navarro [73] respectively.…”

On a Conjecture of G. Malle and G. Navarro on Nilpotent Blocks

“…Since χ i and χ 0 both have p-height 0, we have H p (λ (w) i ) = H p (λ (w) 0 ). Also, since all the (p)-bars in λ 1 or of type 2, Proposition 2.5 in[4] gives H p ′ (λ(w) i ) ≡ ±H(γ) (mod p) and H p ′ (λ (w) 0 ) ≡ ±H(γ) (mod p). If H(λ and since H(γ) is invertible (mod p), this implies 1 ≡…”