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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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On the Cartan matrix of Mackey algebras
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by Serge Bouc PDF
Trans. Amer. Math. Soc. 363 (2011), 4383-4399 Request permission

Abstract:

Let $k$ be a field of characteristic $p>0$, and let $G$ be a finite group. The first result of this paper is an explicit formula for the determinant of the Cartan matrix of the Mackey algebra $\mu _k(G)$ of $G$ over $k$. The second one is a formula for the rank of the Cartan matrix of the cohomological Mackey algebra $co\mu _k(G)$ of $G$ over $k$, and a characterization of the groups $G$ for which this matrix is nonsingular. The third result is a generalization of this rank formula and characterization to blocks of $co\mu _k(G)$: in particular, if $b$ is a block of $kG$, the Cartan matrix of the corresponding block $co\mu _k(b)$ of $co\mu _k(G)$ is nonsingular if and only if $b$ is nilpotent with cyclic defect groups.
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Additional Information
  • Serge Bouc
  • Affiliation: LAMFA - CNRS UMR 6140, Université de Picardie Jules Verne, 33, rue St Leu, 80039 Amiens, France
  • MR Author ID: 207609
  • ORCID: 0000-0003-2330-1845
  • Email: serge.bouc@u-picardie.fr
  • Received by editor(s): October 6, 2009
  • Received by editor(s) in revised form: January 2, 2010
  • Published electronically: March 22, 2011
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 363 (2011), 4383-4399
  • MSC (2010): Primary 18G05, 20C20, 20J06
  • DOI: https://doi.org/10.1090/S0002-9947-2011-05291-8
  • MathSciNet review: 2792992