Foncteurs de Mackey projectifs

Abstract

This work deals with the study of projective Mackey functors. Mackey functors are algebraic structures with operations which behave like induction, restriction and conjugation in group representation theory. These objects have properties which generalize many constructions such as, for example, group cohomology, the Burnside ring or algebraic K-theory of group rings. In the first part we concentrate on extension groups of degree 1 between simple Mackey functors for a group G. The calculation of these groups is a very important tool in determining the Loewy series of projective Mackey functors. We determine extension groups between simple Mackey functors indexed by normal subgroups of G. We next study the conditions under which it is possible to restrict ourselves to that case, and we give methods for calculating extension groups between simple Mackey functors which are not indexed by normal subgroups.We also study the case of extension groups between simple Mackey functors indexed by the same subgroup. In that case, every extension group can be embedded in an extension group between modules over a group algebra, and we describe the image of this embedding. In particular, we determine every extension group for simple Mackey functors for a p-group and for a group which has a normal p-Sylow subgroup. Next, we focus on extension groups of higher degree between simple Mackey functors for a group with a p-Sylow subgroup of order p. We calculate explicitly the minimal projective resolution of a simple Mackey functor for the group Cp ⋊ Ce, where e divides p - 1 and Ce acts faithfully on Cp. This allows us to prove that every simple Mackey functor for a group whose order is not divisible by p2 has a periodic (or finite) minimal projective resolution. Next, we examine the socle of a projective Mackey functor for a p-group P. We prove that simple subfunctors of a projective functor indexed by a subgroup H of P are indexed by normalizers in H of subgroups of H. In particular, this implies that in the case where P is abelian, every simple subfunctor of our projective functor is indexed by H. We then study the socle of a specific projective Mackey functor, namely the Burnside functor BP, and we focus on the case where P is abelian. In particular, we calculate it in the case of a cyclic p-group, an abelian p-group of rank 2 and an elementary abelian p-group of rank 3. This enables us to determine the socle of an indecomposable projective Mackey functor indexed by a subgroup of P isomorphic to one of the previous groups. We end this work by providing an explicit formula for the Cartan coefficients of the Mackey functors for a p-group.