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

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

On the Cartan matrix of Mackey algebras

Let k be a field of characteristic p>0, and G be a finite group. The first result of this paper is an explicit formula for the determinant of the Cartan matrix of the Mackey algebra mu_k(G) of G over k. The second one is a formula for the rank of the Cartan matrix of the cohomological Mackey algebra comu_k(G) of G over k, and a characterization of the groups G for which this matrix is non singular. The third result is a generalization of this rank formula and characterization to blocks of comu_k(G) : in particular, if b is a block of kG, the Cartan matrix of the corresponding block comu_k(b) of comu_k(G) is non singular if and only if b is nilpotent with cyclic defect groups

Rational pp-biset functors

In this paper, I give several characterizations of {\em rational biset functors over $p$-groups}, which are independent of the knowledge of genetic bases for $p$-groups. I also introduce a construction of new biset functors from known ones, which is similar to the Yoneda construction for representable functors, and to the Dress construction for Mackey functors, and I show that this construction preserves the class of rational $p$-biset functors.\par This leads to a characterization of rational $p$-biset functors as additive functors from a specific quotient category of the biset category to abelian groups. Finally, I give a description of the largest rational quotient of the Burnside $p$-biset functor : when $p$ is odd, this is simply the functor R_\Q of rational representations, but when $p=2$, it is a non split extension of R_\Q by a specific uniserial functor, which happens to be closely related to the functor of units of the Burnside ring