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arXiv · 0706.4205

The extended Burnside ring and module categories

Andrew Rose
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Abstract

In this note an `extended Burnside ring' is defined, generated by classes of semisimple module categories over Rep(G) with quasifibre functors. Here G is a finite group and representations are taken over an algebraically closed field of characteristic 0. It is shown that this is equivalent to a ring generated by centrally extended G-sets and hence the name. Ring homomorphisms into the multiplicative group of the field are computed with an explicit formula and tables of these homomorphisms are given for the groups S_4 and S_5 which are of particular interest in the context of reductive algebraic groups. ----- L'anneau de Burnside étendu et catégories de modules. Dans cette note un `Anneau de Burnside étendu' est défini, generé par des classes de catégories de modules semisimples sur Rep(G) avec des foncteurs quasifibres. Ici G est un groupe fini, et des représentations sont prises sur un corps algébriquement clos de caractéristique nulle. Il est demontré que ceci équivaut à un anneau generé par des G-ensembles centralement étendus, d'où le nom. Des homomorphismes d'anneau dans le groupe multiplicatif du corps sont computées avec une formule explicite et des tableaux de ces homomorphismes sont fournis pour les groupes S_4 et S_5 qui sont d'un intérêt particulier dans le contexte de groupes algébriques réductifs.

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Andrew Rose. 2011-10-27. The extended Burnside ring and module categories. https://arxiv.org/abs/0706.4205

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