Simple section biset functors

Abstract

Let G and H be finite groups and k be a commutative unitary ring. The Burnside group B(G,H) is the Grothendieck group of the category of finite (G,H)-bisets. The biset category kC of finite groups is the category defined over finite groups, whose morphism sets are given by the kB(G,H) groups. A biset functor defined on kC, with values in k-Mod is a k-linear functor from kC to the category of k-Mod. The remarkable results as the evaluation of the Dade group of endopermutation modules of a p- group and finding the unit group of the Burnside ring of a p- group are done using the theory of biset functors. Looking for ring objects in the category of biset functors one gets a more sophisticated structure, which is called a Green Biset Functor. Serge Bouc introduced the slice Burnside ring and the section Burnside ring for a finite group G. He also showed that these two rings have a natural structure of a Green Biset Functor. In our work we classify simple modules over the section Burnside ring of G using the approach of the paper Fibered Biset Functors by Robert Boltje and Olcay Coşkun.