Green correspondence on centric Mackey functors over fusion systems

Abstract

In this paper we give a definition of (centric) Mackey functor over a fusion system (Definitions 2.26 and 2.29) 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 in [1]) acts on any centric Mackey functor (Proposition 2.43). We also prove that the Green correspondence holds for centric Mackey functors over fusion systems (Theorem 4.37). As a mean to prove this we introduce a notion of relative projectivity for centric Mackey functors over fusion systems (Definition 3.1) and provide a decomposition of a particular product in O(Fc)⊔ (Definition 2.12) in terms of the product in O((NF)c)⊔ (Theorem 4.27).