Details
| Original language | English |
|---|---|
| Pages (from-to) | 345-362 |
| Number of pages | 18 |
| Journal | Journal of algebra |
| Volume | 337 |
| Issue number | 1 |
| Publication status | Published - 1 Jul 2011 |
| Externally published | Yes |
Abstract
This paper continues Sambale (2011) [28]. We show that the methods developed there also work for odd primes. In particular we prove Brauer's k(B)-conjecture for defect groups which contain a central, cyclic subgroup of index at most 9. As a consequence, the k(B)-conjecture holds for 3-blocks of defect at most 3. In the second part of the paper we illustrate the limits of our methods by considering an example. Then we use the work of Kessar, Koshitani and Linckelmann [13] (and thus the classification) to show that the k(B)-conjecture is satisfied for 2-blocks of defect 5 except for the extraspecial defect group D8*D8. As a byproduct we also obtain the block invariants of 2-blocks with minimal nonmetacyclic defect groups. Some proofs rely on computer computations with GAP (The GAP Group, 2008 [10]).
Keywords
- Brauer's k(B)-conjecture, Cartan matrices, Minimal nonmetacyclic defect groups
ASJC Scopus subject areas
- Mathematics(all)
- Algebra and Number Theory
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In: Journal of algebra, Vol. 337, No. 1, 01.07.2011, p. 345-362.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Cartan matrices and Brauer's k(B)-conjecture II
AU - Sambale, Benjamin
N1 - Funding Information: I thank Shigeo Koshitani for showing me [31]. I am also very grateful to the referee for greatly simplifying the proof of Lemma 1. This work was partly supported by the “Deutsche Forschungsge-meinschaft”.
PY - 2011/7/1
Y1 - 2011/7/1
N2 - This paper continues Sambale (2011) [28]. We show that the methods developed there also work for odd primes. In particular we prove Brauer's k(B)-conjecture for defect groups which contain a central, cyclic subgroup of index at most 9. As a consequence, the k(B)-conjecture holds for 3-blocks of defect at most 3. In the second part of the paper we illustrate the limits of our methods by considering an example. Then we use the work of Kessar, Koshitani and Linckelmann [13] (and thus the classification) to show that the k(B)-conjecture is satisfied for 2-blocks of defect 5 except for the extraspecial defect group D8*D8. As a byproduct we also obtain the block invariants of 2-blocks with minimal nonmetacyclic defect groups. Some proofs rely on computer computations with GAP (The GAP Group, 2008 [10]).
AB - This paper continues Sambale (2011) [28]. We show that the methods developed there also work for odd primes. In particular we prove Brauer's k(B)-conjecture for defect groups which contain a central, cyclic subgroup of index at most 9. As a consequence, the k(B)-conjecture holds for 3-blocks of defect at most 3. In the second part of the paper we illustrate the limits of our methods by considering an example. Then we use the work of Kessar, Koshitani and Linckelmann [13] (and thus the classification) to show that the k(B)-conjecture is satisfied for 2-blocks of defect 5 except for the extraspecial defect group D8*D8. As a byproduct we also obtain the block invariants of 2-blocks with minimal nonmetacyclic defect groups. Some proofs rely on computer computations with GAP (The GAP Group, 2008 [10]).
KW - Brauer's k(B)-conjecture
KW - Cartan matrices
KW - Minimal nonmetacyclic defect groups
UR - http://www.scopus.com/inward/record.url?scp=79956275101&partnerID=8YFLogxK
U2 - 10.1016/j.jalgebra.2011.03.023
DO - 10.1016/j.jalgebra.2011.03.023
M3 - Article
AN - SCOPUS:79956275101
VL - 337
SP - 345
EP - 362
JO - Journal of algebra
JF - Journal of algebra
SN - 0021-8693
IS - 1
ER -