The inductive Alperin–McKay and Blockwise Alperin Weight conditions for blocks with cyclic defect groups and odd primes

Abstract

We verify the inductive blockwise Alperin weight (BAW) and the inductive Alperin-McKay (AM) conditions introduced by the second author for blocks of finite quasisimple groups with cyclic defect groups. Furthermore we establish a criterion that describes conditions under which the inductive AM condition for blocks with abelian defect groups implies the inductive BAW condition for those blocks.

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.…”

Clifford theory of characters in induced blocks

“…This situation turns out in [KS13] to be important in checking the inductive Alperin-McKay (inductive AM) and inductive Blockwise-Alperin-Weight (inductive BAW) conditions for blocks with cyclic defect groups.…”

“…As a consequence we obtain a sort of generalization of the Theorem of . to [KS13]. Those inductive conditions are defined in the context of the reduction theorems of the Alperin-McKay and Alperin weight conjecture.…”

On the inductive blockwise Alperin weight condition for classical groups

“…4.1], there is also a version treating only blocks with defect groups involved in certain sets of ℓ-groups [54,Def. 5.17], or a version handling single blocks [36,Def. 3.2].…”