Mathematics – Representation Theory
Scientific paper
2007-06-28
Mathematics
Representation Theory
This paper has been superceded by 1110.4722
Scientific paper
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 \'etendu et cat\'egories de modules. Dans cette note un `Anneau de Burnside \'etendu' est d\'efini, gener\'e par des classes de cat\'egories de modules semisimples sur Rep(G) avec des foncteurs quasifibres. Ici G est un groupe fini, et des repr\'esentations sont prises sur un corps alg\'ebriquement clos de caract\'eristique nulle. Il est demontr\'e que ceci \'equivaut \`a un anneau gener\'e par des G-ensembles centralement \'etendus, d'o\`u le nom. Des homomorphismes d'anneau dans le groupe multiplicatif du corps sont comput\'ees avec une formule explicite et des tableaux de ces homomorphismes sont fournis pour les groupes S_4 et S_5 qui sont d'un int\'er\^et particulier dans le contexte de groupes alg\'ebriques r\'eductifs.
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