On the Sum of Squares of Irreducible Brauer Character Degrees in Blocks

TL;DR

This paper investigates a weakened form of the Holm--Willems Local Conjecture in block theory, focusing on cases with abelian defect groups and providing complete proofs for the case p=2.

Contribution

It reduces the conjecture to quasi-simple groups under abelian defect group assumptions and offers complete proofs for p=2.

Findings

Confirmed the conjecture for p=2 cases.

Reduced the problem to quasi-simple groups with abelian defect groups.

Provided partial results towards the conjecture.

Abstract

We study a weakened version of the Holm--Willems Local Conjecture. The problem is reduced to quasi-simple groups under the assumption that the defect group is abelian. Complete proofs are provided in the case \(p = 2\).