Some results on a variant of the Green correspondence with applications to Alperin's weight conjecture for blocks

Abstract

Alperin’s Weight Conjecture for Blocks (AWCFB) suggests a currently unknown structure in Finite Group Modular Representation Theory. The Green Correspondence fails AWCFB for blocks of p-solvable groups. An important result of L. Barker uses a variant of the Green Correspondence to prove AWFCB for blocks of p-solvable groups. This variant suggests the possibility of further solutions to AWCFB. Let G be a finite group, let N be a normal subgroup of G and let b be a block of N that is covered by the block B of G. Our main results demonstrate that: for B it suffices to obtain a required variant of the Green Correspondence for the block of the stabilizer of b corresponding to B and if b is of defect zero, then it suffices to reduce to the group G/N . L. Barker’s proof of the AWCFB for p-solvable groups is an immediate consequence of our results. Mathematics Subject Classification: 20C20