Cartan matrices and Brauer's k(B)-Conjecture III

TL;DR

This paper establishes bounds involving Cartan matrices for blocks of finite groups and proves Brauer's k(B)-Conjecture for certain cases, including blocks with abelian defect groups and small l(B).

Contribution

It provides new bounds relating Cartan matrices to irreducible characters and proves Brauer's k(B)-Conjecture for blocks with abelian defect groups under specific conditions.

Findings

Proves bounds on k(B) using Cartan matrix determinants.

Confirms Brauer's k(B)-Conjecture for blocks with abelian defect groups and certain inertial quotients.

Validates the conjecture for blocks with l(B) ≤ 3.

Abstract

For a block $B$ of a finite group we prove that $k(B)\le(\det C-1)/l(B)+l(B)\le\det C$ where $k(B)$ (respectively $l(B)$) is the number of irreducible ordinary (respectively Brauer) characters of $B$, and $C$ is the Cartan matrix of $B$. As an application, we show that Brauer's $k(B)$-Conjecture holds for every block with abelian defect group $D$ and inertial quotient $T$ provided there exists an element $u\in D$ such that $C_T(u)$ acts freely on $D/<u>$. This gives a new proof of Brauer's Conjecture for abelian defect groups of rank at most $2$. We also prove the conjecture in case $l(B)\le 3$.