A progenerator for representations of $ {\mathbf{SL}}_{n}({\mathbb{F}}_{q})$ in transverse characteristic

Cédric Bonnafé

doi:10.1016/j.crma.2011.06.008

Algebra/Group Theory

A progenerator for representations of

{SL}_{n} (F_{q})

in transverse characteristic

Presented by: the Editorial Board

Cédric Bonnafé ¹

¹ CNRS – UMR 5149, Institut de mathématiques et de modélisation de Montpellier, Université Montpellier 2, place Eugène-Bataillon, 34095 Montpellier cedex, France

Comptes Rendus. Mathématique, Volume 349 (2011) no. 13-14, pp. 731-733

Abstract (Original language)
Résumé

Let $G = {GL}_{n} (F_{q})$ , ${SL}_{n} (F_{q})$ or ${PGL}_{n} (F_{q})$ , where q is a power of some prime number p, let U denote a Sylow p-subgroup of G and let R be a commutative ring in which p is invertible. Let $D (U)$ denote the derived subgroup of U and let $e = \frac{1}{| D (U) |} \sum_{u \in D (U)} u$ . The aim of this Note is to prove that the R-algebras RG and $e R G e$ are Morita equivalent (through the natural functor RG-mod → $e R G e$ -mod, $M \mapsto e M$ ).

Soit $G = {GL}_{n} (F_{q})$ , ${SL}_{n} (F_{q})$ ou ${PGL}_{n} (F_{q})$ , où q est une puissance dʼun nombre premier p, soit U un p-sous-groupe de Sylow de G et soit R un anneau commutatif dans lequel p est inversible. Soit $D (U)$ le groupe dérivé de U et soit $e = \frac{1}{| D (U) |} \sum_{u \in D (U)} u$ . Le but de cette Note est de montrer que les R-algèbres RG et $e R G e$ sont Morita équivalentes (à travers le foncteur naturel RG-mod → $e R G e$ -mod, $M \mapsto e M$ ).

Received: 2011-03-27
Accepted: 2011-06-07
Published online: 2011-07-01

DOI: 10.1016/j.crma.2011.06.008

Author's affiliations:

Cédric Bonnafé ¹

¹ CNRS – UMR 5149, Institut de mathématiques et de modélisation de Montpellier, Université Montpellier 2, place Eugène-Bataillon, 34095 Montpellier cedex, France

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     author = {C\'edric Bonnaf\'e},
     title = {A progenerator for representations of $ {\mathbf{SL}}_{n}({\mathbb{F}}_{q})$ in transverse characteristic},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {731--733},
     year = {2011},
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     doi = {10.1016/j.crma.2011.06.008},
     language = {en},
}

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Cédric Bonnafé. A progenerator for representations of $ {\mathbf{SL}}_{n}({\mathbb{F}}_{q})$ in transverse characteristic. Comptes Rendus. Mathématique, Volume 349 (2011) no. 13-14, pp. 731-733. doi: 10.1016/j.crma.2011.06.008

References
Cited by

[1] C. Bonnafé; R. Rouquier Coxeter orbits and modular representations, Nagoya Math. J., Volume 183 (2006), pp. 1-34

[2] R. Dipper On quotients of Hom-functors and representations of finite general linear groups II, J. Algebra, Volume 209 (1998), pp. 199-269

[3] T.-Y. Lam Lectures on Modules and Rings, Graduate Texts in Mathematics, vol. 189, Springer, 1999 (xxiv+557 pp)

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