Serge Bouc

Biset functors for finite groups

Examples.- General properties.- -Sets and (, )-Bisets.- Biset Functors.- Simple Functors.- Biset functors on replete subcategories.- The Burnside Functor.- Endomorphism Algebras.- The Functor.- Tensor Product and Internal Hom.- p-biset functors.- Rational Representations of -Groups.- -Biset Functors.- Applications.- The Dade Group.

Green functors and G-sets

Mackey functors.- Green functors.- The category associated to a green functor.- The algebra associated to a green functor.- Morita equivalence and relative projectivity.- Construction of green functors.- A morita theory.- Composition.- Adjoint constructions.- Adjunction and green functors.- The simple modules.- Centres.

The functor of units of Burnside rings for p-groups

In this paper, I describe the structure of the biset functor B× sending a p-group P to the group of units of its Burnside ring B(P). In particular, I show that B× is a rational biset functor. It follows that if P is a p-group, the structure of B×(P) can be read from a genetic basis of P: the group B×(P) is an elementary abelian 2-group of rank equal to the number isomorphism classes of rational irreducible representations of P whose type is trivial, cyclic of order 2, or dihedral.

Hochschild Constructions for Green Functors

Abstract Let G be a finite group, and R be a commutative ring. If A is a Green functor for G over R, and Γ is a crossed G-monoid, then the Mackey functor AΓ obtained by the Dress construction has a natural structure of Green functor, and its evaluation AΓ(G) is an R-algebra. This framework involves as special cases the construction of the Hochschild cohomology algebra of the group algebra from the ordinary cohomology functor, and the construction of the crossed Burnside ring from the ordinary Burnside functor. This article presents some properties of those Green functors A Γ, and the functorial relations between the corresponding categories of modules. As a consequence, a general product formula for the algebra A Γ(G) is stated.

Non-additive exact functors and tensor induction for Mackey functors

Introduction Non additive exact functors Permutation Mackey functors Tensor induction for Mackey functors Relations with the functors

Gluing endo-permutation modules

{\mathcal L}_U

A sectional characterization of the Dade group

Direct product of Mackey functors Tensor induction for Green functors Cohomological tensor induction Tensor induction for

ON THE CARTAN MATRIX OF MACKEY ALGEBRAS

p

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

-permutation modules Tensor induction for modules Bibliography.

On the Projective Dimensions of Mackey Functors

Abstract In this paper, it is shown that if p is an odd prime, and if P is a finite p-group, then there exists an exact sequence of abelian groups , where D(P) is the Dade group of P and T(P) is the subgroup of endo-trivial modules. Here is the group of sequences of compatible elements in the Dade groups D(NP (Q)/Q) for non-trivial subgroups Q of P. The poset is the set of elementary abelian subgroups of rank at least 2 of P, ordered by inclusion. The group is the subgroup of consisting of classes of P-invariant 1-cocycles. A key result for the proof that the above sequence is exact is a characterization of elements of 2D(P) by sequences of integers, indexed by sections (T, S) of P such that T/S ≅ (ℤ/pℤ)2, fulfilling certain conditions associated to subquotients of P which are either elementary abelian of rank 3, or extraspecial of order p 3 and exponent p.