Details
| Original language | English |
|---|---|
| Pages (from-to) | 416-427 |
| Number of pages | 12 |
| Journal | Journal of algebra |
| Volume | 331 |
| Issue number | 1 |
| Publication status | Published - 1 Apr 2011 |
| Externally published | Yes |
Abstract
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.
Keywords
- Brauer's k(B)-conjecture, Cartan matrices, Decomposition matrices, Quadratic forms
ASJC Scopus subject areas
- Mathematics(all)
- Algebra and Number Theory
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In: Journal of algebra, Vol. 331, No. 1, 01.04.2011, p. 416-427.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Cartan matrices and Brauer's k(B)-conjecture
AU - Sambale, Benjamin
PY - 2011/4/1
Y1 - 2011/4/1
N2 - 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.
AB - 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.
KW - Brauer's k(B)-conjecture
KW - Cartan matrices
KW - Decomposition matrices
KW - Quadratic forms
UR - http://www.scopus.com/inward/record.url?scp=79952008751&partnerID=8YFLogxK
U2 - 10.1016/j.jalgebra.2010.10.036
DO - 10.1016/j.jalgebra.2010.10.036
M3 - Article
AN - SCOPUS:79952008751
VL - 331
SP - 416
EP - 427
JO - Journal of algebra
JF - Journal of algebra
SN - 0021-8693
IS - 1
ER -