The slice Burnside ring and the section Burnside ring of a finite group

Serge Bouc (1)

1 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée

This paper introduces two new Burnside rings for a finite group $G$, called the slice Burnside ring and the section Burnside ring. They are built as Grothendieck rings of the category of morphisms of $G$-sets, and of Galois morphisms of $G$-sets, respectively. The well known results on the usual Burnside ring, concerning ghost maps, primitive idempotents, and description of the prime spectrum, are extended to these rings. It is also shown that these two rings have a natural structure of Green biset functor. The functorial structure of unit groups of these rings is also discussed.

Mots clés: Burnside ring; Green biset functor; slice; section; unit group

Domaines: Théorie des groupes [math.GR]; Anneaux et algèbres [math.RA]; Catégories et ensembles [math.CT]; K-théorie et homologie [math.KT]

Serge Bouc. The slice Burnside ring and the section Burnside ring of a finite group. Compositio Mathematica, 2012, 148, pp.868-906. ⟨hal-00556438⟩