A criterion for the inductive Alperin weight condition

Abstract

We give a criterion that simplifies the checking of the inductive Alperin weight condition for the remaining open cases of simple groups of Lie type (see Theorem 3.3 below). It is strongly related in form to the criterion of the second author for the inductive McKay conditions (see [Spä12, 2.12]) that has proved very useful. The proof follows from a Clifford theory for weights intrinsically present in the proof of reduction theorems of the Alperin weight conjecture given by Navarro-Tiep and the second author. We also give a related criterion for the inductive blockwise Alperin weight condition (Theorem 4.5).

The Inductive Blockwise Alperin Weight Condition for Type and the Prime

“…Here for every l ∈ L, the linear character λ ′ l ∈ Irr(N X (D)/ N N (D)) is determined by ψ 0 = λ ′ l ψ l 0 . As in the proof of [12,Lemma 3.4], (P * , P ′ * ) gives (G, M, θ) (N G (D), N M (D), θ ′ ). In addition, it can be checked that (P * , P ′ * ) gives (G, M, θ) c (N G (D), N M (D), θ ′ ) immediately.…”

“…Let G G. Now we recall the relationship "covering" for weights between G and G defined in [12, §2] [12], we say that a weight (…”

“…Proof. We will use the criterion for the inductive BAW condition given by Brough and Späth [12] and see [23,Thm. 3.18].…”

“…In [20], the weights of G were classified and the inductive condition of the non-blockwise Alperin weight conjecture was verified for groups of type A. By the criterion given by Brough-Späth [12], the key of the verification of the inductive BAW condition for the simple group G/ Z(G) is to establish an Aut(G)-equivariant bijection between IBr(G) and Alp(G) which preserves blocks.…”

Unitriangular basic sets, Brauer characters and coprime actions

“…Remark 7.4. In view of the criterion from [BS20] of the inductive blockwise Alperin weight condition from [S13] the above shows that, for many blocks B of finite quasi-simple groups G of Lie type, IBr(B) is Aut(G) B -permutation isomorphic to a subset of Irr(G). This is needed in [FLZ21b] and [L21] in the verification of the inductive blockwise Alperin weight condition for types B and C. More generally, with the assumptions of Proposition 7.3 and assuming in addition that G := G F satisfies A(∞) (see Definition 5.1), it is easy to see that IBr(B s ) satisfies assumption (ii) of Theorem 4.9.…”