Representation rings for fusion systems and dimension functions

Abstract

We define the representation ring of a saturated fusion system \(\mathcal F\) as the Grothendieck ring of the semiring of \(\mathcal F\)-stable representations, and study the dimension functions of \(\mathcal F\)-stable representations using the transfer map induced by the characteristic idempotent of \(\mathcal F\). We find a list of conditions for an \(\mathcal F\)-stable super class function to be realized as the dimension function of an \(\mathcal F\)-stable virtual representation. We also give an application of our results to constructions of finite group actions on homotopy spheres.