The strengthened Alperin Weight Conjecture for p-solvable groups

Abstract

The Alperin Weight Conjecture is a well-known conjecture, and it is central to the modern representation theory of finite groups. It is known to be true for many types of finite groups, but it remains open. In particular, the conjecture is known to be true for finite p-solvable groups. In this paper, we prove that a very strong form of the conjecture holds for all finite p-solvable groups. It follows that the strengthening of the Alperin Weight Conjecture that includes Galois automorphisms over the p-adic numbers holds for all finite p-solvable groups. This strengthening of the Alperin Weight Conjecture was suggested in an earlier work of Navarro on the McKay Conjecture. The present paper provides the first direct evidence (beyond the calculation of individual cases and the case of groups of odd order) for this strengthening.