Green correspondence on centric Mackey functor over fusion systems

Abstract

In this paper we give a definition of (centric) Mackey functor over a fusion system which generalizes the notion of Mackey functor over a group. In this context we prove that, given some conditions on a related ring, the centric Burnside ring over a fusion system (as defined by Diaz and Libman) acts on any centric Mackey functor. We also prove that the Green correspondence holds for centric Mackey functors over fusion systems. As a means to prove this we introduce a notion of relative projectivity for centric Mackey functors over fusion systems and provide a decomposition of a particular product in $\mathcal{O}{\left({\mathcal{F}}^c\right)}_{\bigsqcup }$ in terms of the product in $\mathcal{O}{\left(N_{\mathcal{F}}{\left(H\right)}^c\right)}_{\bigsqcup }$.

References in corpus (1)

• Transfer and characteristic idempotents for saturated fusion systems