Green correspondence on centric Mackey functor over fusion systems

TL;DR

This paper extends the concept of Mackey functors to fusion systems, proving the Green correspondence and establishing the action of the centric Burnside ring on these functors, thus generalizing classical representation theory results.

Contribution

It introduces a definition of centric Mackey functors over fusion systems and proves the Green correspondence within this new framework.

Findings

Centric Burnside ring acts on centric Mackey functors under certain conditions.

Green correspondence holds for centric Mackey functors over fusion systems.

Decomposition of products in the orbit category related to 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 $\mathcal{O}\left(\mathcal{F}^c\right)_{\sqcup}$ in terms of the product in $\mathcal{O}\left(N_{\mathcal{F}}\left(H\right)^c\right)_{\sqcup}$.