Weights for $\pi$-partial characters of $\pi$-separable groups

TL;DR

This paper proves an inequality related to $ au$-weights and $ au$-partial characters in $ au$-separable groups, confirming a prediction by Isaacs and Navarro, and provides conditions for their equality, advancing the $ au$-version of the Alperin weight conjecture.

Contribution

It confirms a predicted inequality between $ au$-weights and $ au$-partial characters and offers conditions for their equality, strengthening the $ au$-version of the Alperin weight conjecture.

Findings

Confirmed the inequality between $ au$-weights and $ au$-partial characters.

Provided sufficient conditions for equality of these quantities.

Strengthened the main theorem on the $ au$-version of the Alperin weight conjecture.

Abstract

The aim of this paper is to confirm an inequality predicted by Isaacs and Navarro in 1995, which asserts that for any $\pi'$-subgroup $Q$ of a $\pi$-separable group $G$, the number of $\pi'$-weights of $G$ with $Q$ as the first component always exceeds that of irreducible $\pi$-partial characters of $G$ with $Q$ as their vertex. We also give some sufficient condition to guarantee that these two numbers are equal, and thereby strengthen their main theorem on the $\pi$-version of the Alperin weight conjecture.