Lifts of Brauer characters in characteristic two

TL;DR

This paper investigates the lifting properties of irreducible 2-Brauer characters in solvable groups, establishing conditions for linear Navarro vertices and conjugacy of twisted vertices, and proving a weaker form of Cossey's conjecture in characteristic two.

Contribution

It provides a characterization of lifts with linear Navarro vertices in characteristic two and introduces a new approach to Cossey's conjecture for p=2.

Findings

Lifts of 2-Brauer characters have linear Navarro vertices iff all vertex pairs are linear.

All twisted vertices of such lifts are conjugate under certain conditions.

A weaker form of Cossey's conjecture is proved for p=2, focusing on one vertex at a time.

Abstract

A conjecture raised by Cossey in 2007 asserts that if $G$ is a finite $p$-solvable group and $\varphi$ is an irreducible $p$-Brauer character of $G$ with vertex $Q$, then the number of lifts of $\varphi$ is at most $|Q:Q'|$. This conjecture is now known to be true in several situations for $p$ odd, but there has been little progress for $p$ even. The main obstacle appeared in characteristic two is that all the vertex pairs of a lift are neither linear nor conjugate. In this paper we show that if $\chi$ is a lift of an irreducible $2$-Brauer character in a solvable group, then $\chi$ has a linear Navarro vertex if and only if all the vertex pairs of $\chi$ are linear, and in that case all of the twisted vertices of $\chi$ are conjugate. Our result can also be used to study other lifting problems of Brauer characters in characteristic two. As an application, we prove a weaker form of Cossey's conjecture for $p=2$ "one vertex at a time".