Parameterization of irreducible characters for p-solvable groups

TL;DR

This paper refines the parameterization of irreducible characters for p-solvable groups, establishing a natural bijection that respects group actions, blocks, and defects, advancing understanding of character theory in finite groups.

Contribution

It introduces a refined method to parameterize irreducible characters of p-solvable groups, compatible with automorphisms and block structures, building on previous theoretical frameworks.

Findings

Provides a natural bijection between irreducible characters and inductive weights

Ensures compatibility with outer automorphisms of the group

Preserves blocks and defects in the parameterization

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. In 1992, Everett Dade formulates a refinement of Alperin's conjecture involving ordinary irreducible characters - with their defect - and, in 2000, Geoffrey Robinson proves that the new conjecture holds for p-solvable groups. But this refinement is formulated in terms of a vanishing alternating sum, without giving any possible refinement for the weights. In this note we show that, in the case of the p-solvable finite groups, the method developed in a previous paper [arXiv.org/abs/1005.3748] can be suitably refined to provide, up to the choice of a polarization, a natural bijection - namely compatible with the action of the group of outer automorphisms of G - between the sets of absolutely irreducible characters of G and of G-conjugacy classes of suitable inductive weights, preserving blocks and defects.