Details
| Original language | English |
|---|---|
| Pages (from-to) | 591-603 |
| Number of pages | 13 |
| Journal | Transactions of the American Mathematical Society |
| Volume | 359 |
| Issue number | 2 |
| Publication status | Published - Feb 2007 |
| Externally published | Yes |
Abstract
Recently, a new conjecture on the degrees of the irreducible Brauer characters of a finite group was presented by W. Willems. In this paper we propose a 'local' version of this conjecture for blocks B of finite groups, giving a lower bound for Σψ(1)2 where the sum runs through the set of irreducible Brauer characters of B in terms of invariants of B. A slight reformulation leads to interesting open questions about traces of Cartan matrices of blocks. We show that the local conjecture is true for blocks with one simple module, blocks of p-solvable groups and blocks with cyclic defect groups. It also holds for many further examples of blocks of sporadic groups, symmetric groups or groups of Lie type. Finally we prove that the conjecture is true for blocks of tame representation type.
Keywords
- Block of finite group, Brauer character, Cartan matrix, Perron-Frobenius eigenvalue
ASJC Scopus subject areas
- Mathematics(all)
- Mathematics(all)
- Applied Mathematics
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In: Transactions of the American Mathematical Society, Vol. 359, No. 2, 02.2007, p. 591-603.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - A local conjecture on brauer character degrees of finite groups
AU - Holm, Thorsten
AU - Willems, Wolfgang
PY - 2007/2
Y1 - 2007/2
N2 - Recently, a new conjecture on the degrees of the irreducible Brauer characters of a finite group was presented by W. Willems. In this paper we propose a 'local' version of this conjecture for blocks B of finite groups, giving a lower bound for Σψ(1)2 where the sum runs through the set of irreducible Brauer characters of B in terms of invariants of B. A slight reformulation leads to interesting open questions about traces of Cartan matrices of blocks. We show that the local conjecture is true for blocks with one simple module, blocks of p-solvable groups and blocks with cyclic defect groups. It also holds for many further examples of blocks of sporadic groups, symmetric groups or groups of Lie type. Finally we prove that the conjecture is true for blocks of tame representation type.
AB - Recently, a new conjecture on the degrees of the irreducible Brauer characters of a finite group was presented by W. Willems. In this paper we propose a 'local' version of this conjecture for blocks B of finite groups, giving a lower bound for Σψ(1)2 where the sum runs through the set of irreducible Brauer characters of B in terms of invariants of B. A slight reformulation leads to interesting open questions about traces of Cartan matrices of blocks. We show that the local conjecture is true for blocks with one simple module, blocks of p-solvable groups and blocks with cyclic defect groups. It also holds for many further examples of blocks of sporadic groups, symmetric groups or groups of Lie type. Finally we prove that the conjecture is true for blocks of tame representation type.
KW - Block of finite group
KW - Brauer character
KW - Cartan matrix
KW - Perron-Frobenius eigenvalue
UR - http://www.scopus.com/inward/record.url?scp=77950991131&partnerID=8YFLogxK
U2 - 10.1090/S0002-9947-06-03888-8
DO - 10.1090/S0002-9947-06-03888-8
M3 - Article
AN - SCOPUS:77950991131
VL - 359
SP - 591
EP - 603
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
SN - 0002-9947
IS - 2
ER -