Sur quelques foncteurs de bi-ensembles

Abstract

This thesis is in the context of representation theory of finite groups. More specifically, it studies biset functors. In this thesis, I focus on two biset functors: the Burnside functor and the functor of p-permutation modules. For the Burnside functor we first give a result that characterize some B-groups; B-groups being the essential ingredient in the classification of composition factors of the Burnside functor. The second result compares the Burnside functor and the functor of free modules. Note that the functor of free modules is not a biset functor since the inflation of a free module is not necessarily free. To compare those functors we will work on an adjunction between the category of biset functors and the category of functors that do not have inflation. An aspect of the work done on the functor of p-permutation module is to compare the functor of p-permutation modules and the functor of ordinary representations. On the other hand, because of the classification of p-permutation modules, we try to express the functor o p-permutation modules in terms of the functor of projective modules (which is not a biset functor). We will use an adjunction between the category of biset functors and a category that contains the functor of projective modules.