On the inductive Alperin–McKay condition for simple groups of type A

Abstract

As a sequel to [CS13b], we verify the so-called inductive AM-condition introduced in [Sp12] for simple groups of type A and blocks with maximal defect. This is part of the program set up to verify the Alperin-McKay conjecture through its reduction to a problem on quasi-simple groups (see [Sp13]) but also the missing direction of Brauer's height zero conjecture (see [NS14]).Proving that the bijection in [CS13b] preserves ℓ-blocks is relatively easy (see [CS13a, 7.1]) but the additional conditions required by the inductive AM-condition need a specific work. This is explained in Section 2 below. We first give in Definition 2.1 our version of the inductive AM-condition relative to a single defect subgroup. We then show that it suffices to check it for every defect subgroup to get the inductive AM-condition. In passing we take the opportunity to give an equivalent property that may turn out easier to check than the one in [Sp13]; in particular it uses only ordinary characters, no projective representations. In Section 4, we adapt the criterion introduced in [Sp12] and already used in [CS13b] by incorporating blocks to our situation. Some of the methods make use of The second author has been supported by the Deutsche Forschungsgemeinschaft, SPP 1388 and ERC Advanced Grant 291512.. By the definition of φ this leads to bl Res H J 2 ∩H ( ψ) J 2 = bl Reswe know from [Mu13] that bl Res H J 3 ∩H ( ψ) = bl Res H J 2 ∩H ( ψ) J 3 ∩H and bl Res A J 3 ( χ 3 ) = bl Res A J 2 ( χ 3 ) J 3 .

On the Alperin-McKay conjecture for simple groups of type A

“…Additionally [Mal14,SF14] have dealt with simple groups of types 2 B 2 , 2 G 2 and 2 F 4 . Further results have been established in [CS15,KS16b,KS16a] by considering particular structures of the defect group of the block.…”

Local-global conjectures in the representation theory of finite groups

“…This condition has been verified by Koshitani and Späth [55,56] for all blocks with cyclic defect groups, as well as for groups of Lie type in their defining characteristic and alternating groups at odd primes by Späth [93], while Denoncin [32] proved it for alternating groups at p = 2. Cabanes and Späth [25] show it for blocks of SU n (q) and SL n (q) of maximal defect. For the sporadic groups see the website by Breuer [14].…”

Extensions of characters in type D and the inductive McKay condition, I

“…The assumption on the characters Irr cusp (N ) is very similar to the results [CS17a,Prop. 5.13], [CS17b,Thm. 5.1] and [CS19,5.E]…”

“…on Irr(N H (S) F ) for Sylow Φ d -tori S of (H, F ), where H is a simply-connected group of type different from D l and d is a positive integer. The proof there relies on Theorem 4.3 of[CS17b] and we use here a similar strategy. The following proposition gives the road map for the verification of Assumption 1.8(ii).We set W (φ) = N φ /L for every L ≤ M ≤ L and φ ∈ Irr(M ).…”