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

TL;DR

This paper proves Brauer's k(B)-Conjecture for certain 3-blocks with abelian defect groups, introduces a computer algorithm for constructing isotypies, and discovers new perfect isometries, advancing the understanding of block theory.

Contribution

It develops a novel computer algorithm to construct isotypies and proves Brauer's k(B)-Conjecture for specific classes of 3-blocks, including new perfect isometries.

Findings

Proved Brauer's k(B)-Conjecture for 3-blocks with abelian defect groups of rank ≤ 5.

Developed a computer algorithm to construct isotypies based on Usami and Puig's method.

Discovered new perfect isometries for 5-blocks of defect 2.

Abstract

We prove Brauer's k(B)-Conjecture for the 3-blocks with abelian defect groups of rank at most 5 and for all 3-blocks of defect at most 4. For this purpose we develop a computer algorithm to construct isotypies based on a method of Usami and Puig. This leads further to some previously unknown perfect isometries for the 5-blocks of defect 2. We also investigate basic sets which are compatible under the action of the inertial group.