A ghost algebra of the double Burnside algebra in characteristic zero

For a finite group G, we introduce a multiplication on the Q-vector space with basis SG×G, the set of subgroups of G×G. The resulting Q-algebra can be considered as a ghost algebra for the double Burnside ring B(G,G) in the sense that the mark homomorphism from B(G,G) to is a ring homomorphism. Our approach interprets QB(G,G) as an algebra eAe, where A is a twisted monoid algebra and e is an idempotent in A. The monoid underlying the algebra A is again equal to SG×G with multiplication given by composition of relations (when a subgroup of G×G is interpreted as a relation between G and G). The algebras A and are isomorphic via Möbius inversion in the poset SG×G. As an application we improve results by Bouc on the parametrization of simple modules of QB(G,G) and also of simple biset functors, by using results of Linckelmann and Stolorz on the parametrization of simple modules of finite category algebras. Finally, in the case where G is a cyclic group of order n, we give an explicit isomorphism between QB(G,G) and a direct product of matrix rings over group algebras of the automorphism groups of cyclic groups of order k, where k divides n.