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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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A local conjecture on Brauer character degrees of finite groups
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by Thorsten Holm and Wolfgang Willems PDF
Trans. Amer. Math. Soc. 359 (2007), 591-603 Request permission

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
  • Email: tholm@maths.leeds.ac.uk
  • Wolfgang Willems
  • Affiliation: Otto-von-Guericke-Universität, Institut für Algebra und Geometrie, Postfach 4120, 39016 Magdeburg, Germany
  • Email: wolfgang.willems@mathematik.uni-magdeburg.de
  • Received by editor(s): April 25, 2004
  • Received by editor(s) in revised form: October 28, 2004
  • Published electronically: July 21, 2006
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 359 (2007), 591-603
  • MSC (2000): Primary 20C20; Secondary 15A18, 15A36, 16G60, 20C05
  • DOI: https://doi.org/10.1090/S0002-9947-06-03888-8
  • MathSciNet review: 2255187