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Transactions of the American Mathematical Society

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

 
 

 

On the Cartan matrix of Mackey algebras


Author: Serge Bouc
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
Published electronically: March 22, 2011
MathSciNet review: 2792992
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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

Keywords: Cartan matrix, cohomological, Mackey functor, block
Received by editor(s): October 6, 2009
Received by editor(s) in revised form: January 2, 2010
Published electronically: March 22, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.