## On the Cartan matrix of Mackey algebras

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- 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.## References

- David Benson,
*Modular representation theory: new trends and methods*, Lecture Notes in Mathematics, vol. 1081, Springer-Verlag, Berlin, 1984. MR**765858** - D. J. Benson,
*Representations and cohomology. I*, Cambridge Studies in Advanced Mathematics, vol. 30, Cambridge University Press, Cambridge, 1991. Basic representation theory of finite groups and associative algebras. MR**1110581** - S. Bouc,
*Résolutions de foncteurs de Mackey*, Group representations: cohomology, group actions and topology (Seattle, WA, 1996) Proc. Sympos. Pure Math., vol. 63, Amer. Math. Soc., Providence, RI, 1998, pp. 31–83 (French, with English summary). MR**1603131**, DOI 10.1090/pspum/063/1603131 - Serge Bouc,
*The $p$-blocks of the Mackey algebra*, Algebr. Represent. Theory**6**(2003), no. 5, 515–543. MR**2026725**, DOI 10.1023/B:ALGE.0000006493.77655.e5 - Serge Bouc and Jacques Thévenaz,
*The primitive idempotents of the $p$-permutation ring*, J. Algebra**323**(2010), no. 10, 2905–2915. MR**2609181**, DOI 10.1016/j.jalgebra.2009.11.036 - R. Brauer and C. Nesbitt,
*On the modular characters of groups*, Ann. of Math. (2)**42**(1941), 556–590. MR**4042**, DOI 10.2307/1968918 - Michel Broué,
*On Scott modules and $p$-permutation modules: an approach through the Brauer morphism*, Proc. Amer. Math. Soc.**93**(1985), no. 3, 401–408. MR**773988**, DOI 10.1090/S0002-9939-1985-0773988-9 - M. Broué,
*Rickard equivalences and block theory*, Groups ’93 Galway/St. Andrews, Vol. 1 (Galway, 1993) London Math. Soc. Lecture Note Ser., vol. 211, Cambridge Univ. Press, Cambridge, 1995, pp. 58–79. MR**1342782**, DOI 10.1017/CBO9780511629280.009 - Michel Broué and Lluís Puig,
*A Frobenius theorem for blocks*, Invent. Math.**56**(1980), no. 2, 117–128. MR**558864**, DOI 10.1007/BF01392547 - Walter Feit,
*The representation theory of finite groups*, North-Holland Mathematical Library, vol. 25, North-Holland Publishing Co., Amsterdam-New York, 1982. MR**661045** - Daniel Gorenstein,
*Finite groups*, Harper & Row, Publishers, New York-London, 1968. MR**0231903** - M. Nicollerat.
*Foncteurs de Mackey projectifs*. Ph.D. thesis, Ecole Polytechnique Fédérale de Lausanne, 2008. - Jacques Thévenaz and Peter Webb,
*The structure of Mackey functors*, Trans. Amer. Math. Soc.**347**(1995), no. 6, 1865–1961. MR**1261590**, DOI 10.1090/S0002-9947-1995-1261590-5

## 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