A ghost ring for the left-free double Burnside ring and an application to fusion systems

For a finite group G, we define a ghost ring and a mark homomorphism for the double Burnside ring B◁(G,G) of left-free (G,G)-bisets. In analogy to the case of the Burnside ring B(G), the ghost ring has a much simpler ring structure, and after tensoring with Q one obtains an isomorphism of Q-algebras. As an application of a key lemma, we obtain a very general formula for the Brauer construction applied to a tensor product of two p-permutation bimodules M and N in terms of Brauer constructions of the bimodules M and N. Over a field of characteristic 0 we determine the simple modules of the left-free double Burnside algebra and prove semisimplicity results for the bifree double Burnside algebra. These results carry over to results about biset-functor categories. Finally, we apply the ghost ring and mark homomorphism to fusion systems on a finite p-group. We extend a remarkable bijection, due to Ragnarsson and Stancu, between saturated fusion systems and certain idempotents of the bifree double Burnside algebra over Z(p) to a bijection between all fusion systems and a larger set of idempotents in the bifree double Burnside algebra over Q.