On Willems' conjecture on Brauer character degrees

TL;DR

This paper proves Willems' conjecture for the prime 2, establishing a lower bound for the sum of squares of degrees of irreducible Brauer characters in finite groups, and explores related properties of quasi-simple groups.

Contribution

It confirms Willems' conjecture for p=2 using recent reductions and extends understanding of Brauer character degrees and regular semisimple classes in finite groups.

Findings

Willems' conjecture holds for p=2.

Lower bounds for regular semisimple conjugacy classes in finite groups of Lie type.

Verification of Tong-Viet's conditions for certain quasi-simple groups.

Abstract

In 2005 Wolfgang Willems put forward a conjecture proposing a lower bound for the sum of squares of the degrees of the irreducible $p$-Brauer characters of a finite group $G$. We prove this conjecture for the prime $p=2$. For this we rely on the recent reduction of Willems' conjecture to a question on quasi-simple groups by Tong-Viet. We also verify the conditions of Tong-Viet for certain families of finite quasi-simple groups and odd primes. On the way we obtain lower bounds for the number of regular semisimple conjugacy classes in finite groups of Lie type.