Cartan matrices and Brauer's \(k(B)\)-conjecture. II.
This paper continues the author's earlier work in part I [\textit{B. Sambale}, J. Algebra 331, No. 1, 416-427 (2011; Zbl 1230.20010)] and uses this to obtain some more results on Brauer's \(k(B)\)-conjecture. In particular, this conjecture is proved if the defect group contains a central, cyclic subgroup of index at most 9. As a consequence, the conjecture holds for 3-blocks of defect at most 3. It is also shown that the \(k(B)\)-conjecture holds for 2-blocks of defect 5 except possibly the central product of \(D_8\) by itself. (This work ultimately depends on CFSG.) There is also a discussion of an example illustrating the limits of the methods used in the paper. Some proofs are computational in nature and depend on calculations performed with GAP.
numbers of irreducible Brauer characters
numbers of ordinary irreducible characters
scientific article; zbMATH DE number 5996447
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