Mackey functors and sharpness for fusion systems

TL;DR

This paper develops Mackey functors for fusion systems, introduces an acyclicity condition, and extends Dwyer's sharpness results to certain exotic fusion systems with abelian subgroups of index p.

Contribution

It provides a foundational framework for Mackey functors in fusion systems and generalizes sharpness results to new classes of exotic fusion systems.

Findings

Established a parametrization of simple Mackey functors

Extended Dwyer's sharpness results to exotic fusion systems

Provided explicit descriptions of Mackey functors in this context

Abstract

We develop the fundamentals of Mackey functors in the setup of fusion systems including an acyclicity condition as well as a parametrization and an explicit description of simple Mackey functors. Using this machinery we extend Dwyer's sharpness results to exotic fusion systems $\mathcal{F}$ on a finite $p$-group $S$ with an abelian subgroup of index $p$.