Authors: 
Robert Boltje and Susanne Danz

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

Abstract:
For a finite group $G$, we define a ghost ring and a mark 
homomorphism for the double Burnside ring 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 
$\QQ$ one obtains an isomorphism of $\QQ$-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 $\ZZ_{(p)}$, to a bijection between all fusion
systems and a larger set of idempotents in the bifree double Burnside algebra over $\QQ$.

Status:
Submitted for publication