Title
Fusion systems and the double Burnside ring

Abstract

Saturated fusion systems were introduced by Puig as a generalization  
of the $p$-local structure of a finite group or of a block algebra of  
a finite group. Broto, Levi and Oliver introduced the notion of  
characteristic biset associated to a saturated fusion system. This  
biset is not unique but Ragnarsson proved that there is a unique  
characteristic idempotent in the p-completed double Burnside ring  
associated to a saturated fusion system.

In this talk, based o a joint work with Kari Ragnarsson, I will give a
characterization of saturated fusion systems on a $p$-group $S$ in
terms of idempotents in the $p$-local double Burnside ring of $S$ that
satisfy a Frobenius reciprocity relation. This helps us to reformulate
fusion-theoretic phenomena in the language of idempotents and give  
some applications in stable homotopy.