Résume | Given a finite group $G$, the Grothendieck group of the category of finite $(G,G)$-bisets carries a ring structure induced by the tensor product of bisets; the resulting ring $B(G,G)$, called the double Burnside ring of $G$, has in recent years become important for modular representation theory of finite groups, the theory of fusion systems, and topology.
Unlike the classical Burnside ring, the double Burnside ring is in general non-commutative, and its multiplication rather involved. In this talk I will consider the ring $B(G,G)$, its subrings generated by left-free and bifree bisets, as well as the corresponding algebras over fields of characteristic 0. I will present non-commutative analogues of the classical mark homomorphism translating the multiplicative structures of these rings and algebras into more transparent ones. As a first application I will then discuss some questions concerning semisimplicity, structure of simple modules, and possible gradings of double Burnside algebras. This is joint work with Robert Boltje. |