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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
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

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.

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