On the Cartan matrix of Mackey algebras

Bouc, Serge

doi:10.1090/S0002-9947-2011-05291-8

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.

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