Weight parameterization of simple modules for p-solvable groups

TL;DR

This paper establishes a natural bijection between simple modules and weights for p-solvable groups, confirming a key conjecture and relating multiplicity modules, with connections to Navarro's work.

Contribution

It introduces a compatible bijection between simple modules and weights for p-solvable groups and relates multiplicity modules, advancing understanding of Alperin's weight conjecture.

Findings

Established a natural bijection compatible with automorphisms.

Connected multiplicity modules of simple modules and weights.

Showed Navarro's bijection coincides with ours for odd order groups.

Abstract

The weights for a finite group G with respect to a prime number p where introduced by Jon Alperin, in order to formulate his celebrated conjecture affirming that that the number of G-conjugacy classes of weights of G coincides with the number of isomorphism classes of simple kG-modules, where k is an algebraically closed field of characteristic p. Thirty years ago, Tetsuro Okuyama already proved that in the class of p-solvable groups this conjecture holds. In this paper, for the p-solvable groups, on the one hand we exhibit a natural bijection - namely compatible with the action of the group of outer automorphisms of G - between the sets of isomorphism classes of simple G-modules M and of G-conjugacy classes of weights (R,Y), up to the choice of a polarization. On the other hand, we determine the relationship between a multiplicity module of M and Y. In an Appendix, we show that the bijection defined by Gabriel Navarro for the groups of odd order coincides with our bijection for a particular choice of the polarization.