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A local conjecture on Brauer character degrees of finite groups

Authors: Thorsten Holm and Wolfgang Willems
Journal: Trans. Amer. Math. Soc. 359 (2007), 591-603
MSC (2000): Primary 20C20; Secondary 15A18, 15A36, 16G60, 20C05
Published electronically: July 21, 2006
MathSciNet review: 2255187
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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 $ \sum \varphi(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.

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Additional Information

Thorsten Holm
Affiliation: Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom
Address at time of publication: Institut für Algebra und Geometrie, Otto-von-Guericke-Universität, Postfach 4120, 39016 Magdeburg, Germany

Wolfgang Willems
Affiliation: Otto-von-Guericke-Universität, Institut für Algebra und Geometrie, Postfach 4120, 39016 Magdeburg, Germany

Keywords: Brauer character, block of finite group, Cartan matrix, Perron-Frobenius eigenvalue
Received by editor(s): April 25, 2004
Received by editor(s) in revised form: October 28, 2004
Published electronically: July 21, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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