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For a finite group $G$, we define a ghost ring and a mark homomorphism forthe double Burnside ring of left-free $(G,G)$-bisets. In analogy to the case ofthe Burnside ring $B(G)$, the ghost ring has a much simpler ring structure, andafter tensoring with $\QQ$ one obtains an isomorphism of $\QQ$-algebras. As anapplication of a key lemma, we obtain a very general formula for the Brauerconstruction 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 afield of characteristic 0 we determine the simple modules of the left-freedouble Burnside algebra and prove semisimplicity results for the bifree doubleBurnside algebra. These results carry over to results about biset-functorcategories. Finally, we apply the ghost ring and mark homomorphism to fusionsystems on a finite $p$-group. We extend a remarkable bijection, due toRagnarsson and Stancu, between saturated fusion systems and certain idempotentsof the bifree double Burnside algebra over $\ZZ_{(p)}$, to a bijection betweenall fusion systems and a larger set of idempotents in the bifree doubleBurnside algebra over $\QQ.$

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