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

It is well known that the Cartan matrix of a block of a finite group cannot be arranged as a direct sum of smaller matrices. In this paper we address the question if this remains true for equivalent matrices. The motivation for this question comes from the work of Külshammer and Wada (2002) [10], which contains certain bounds for the number of ordinary characters in terms of Cartan invariants. As an application we prove such a bound in the special case, where the determinant of the Cartan matrix coincides with the order of the defect group. In the second part of the paper we show that Brauer's k(B)-conjecture holds for 2-blocks under some restrictions on the defect group. For example, the k(B)-conjecture holds for 2-blocks if the corresponding defect group is a central extension of a metacyclic group by a cyclic group. The same is true if the defect group contains a central cyclic subgroup of index 8. In particular the k(B)-conjecture holds for 2-blocks with defect at most 4. The paper is a part of the author's PhD thesis.