## On the Cartan matrix of Mackey algebras

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- by Serge Bouc
- Trans. Amer. Math. Soc.
**363**(2011), 4383-4399 - DOI: https://doi.org/10.1090/S0002-9947-2011-05291-8
- Published electronically: March 22, 2011
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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.## References

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## Bibliographic 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