On a minimal counterexample to Brauer's $k(B)$-conjecture

TL;DR

This paper investigates Brauer's $k(B)$-conjecture for finite quasi-simple groups, showing that certain blocks do not serve as minimal counterexamples, and precisely determining character counts in unipotent blocks for classical groups at odd primes.

Contribution

It proves that for primes $p eq 3$, blocks of finite quasi-simple groups are not minimal counterexamples to Brauer's $k(B)$-conjecture, and calculates character numbers in unipotent blocks.

Findings

Blocks of finite quasi-simple groups are not minimal counterexamples for $p eq 3$.

Principal 3-blocks do not provide minimal counterexamples.

Number of irreducible characters in unipotent blocks of classical groups is explicitly determined.

Abstract

We study Brauer's long-standing $k(B)$-conjecture on the number of characters in $p$-blocks for finite quasi-simple groups and show that their blocks do not occur as a minimal counterexample for $p\ge5$ nor in the case of abelian defect. For $p=3$ we obtain that the principal 3-blocks do not provide minimal counterexamples. We also determine the precise number of irreducible characters in unipotent blocks of classical groups for odd primes.