Cartan matrices and Brauer's k(B)-conjecture

TL;DR

This paper proves Brauer's k(B)-conjecture for certain defect groups, including central extensions of metacyclic 2-groups and those with specific cyclic subgroups, advancing understanding of block invariants.

Contribution

It establishes the validity of Brauer's k(B)-conjecture for new classes of defect groups, particularly those with minimal nonmetacyclic structures.

Findings

Brauer's k(B)-conjecture holds for defect groups that are central extensions of metacyclic 2-groups.

The conjecture also holds for defect groups with a central cyclic subgroup of index at most 9.

Block invariants of 2-blocks with minimal nonmetacyclic defect groups are determined.

Abstract

We show (among other things) that Brauer's k(B)-conjecture holds for defect groups with are central extensions of metacyclic 2-groups by cyclic groups. The same holds for defect groups which contain a central cyclic subgroup of index at most 9. In particular the k(B)-conjecture holds for 2-blocks of defect at most 4 and 3-blocks of defect at most 3. As a byproduct we obtain the block invariants of 2-blocks with minimal nonmetacyclic defect groups.