Idempotents of double Burnside algebras, L-enriched bisets, and decomposition of p-biset functors

Let R be a (unital) commutative ring, andG be a finite group with order invertible in R. We introduce new idempotents εT,S in the double Burnside algebra RB(G,G) of G over R, indexed by conjugacy classes of minimal sections (T, S) of G (i.e. sections such that S ≤ Φ(T )). These idempotents are orthogonal, and their sum is equal to the identity. It follows that for any biset functor F over R, the evaluation F (G) splits as a direct sum of specific R-modules indexed by minimal sections of G, up to conjugation. The restriction of these constructions to the biset category of p-groups, where p is a prime number invertible in R, leads to a decomposition of the category of p-biset functors over R as a direct product of categories FL indexed by atoric p-groups L up to isomorphism. We next introduce the notions of L-enriched biset and L-enriched biset functor for an arbitrary finite group L, and show that for an atoric pgroup L, the category FL is equivalent to the category of L-enriched biset functors defined over elementary abelian p-groups. Finally, the notion of vertex of an indecomposable p-biset functor is introduced (when p ∈ R×), and when R is a field of characteristic different from p, the objects of the category FL are characterized in terms of vertices of their composition factors. AMS subject classification: 18B99, 19A22, 20J15