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.

22 pages. Corrections and adjustments. Theorem 7.5 added to introduction as Theorem C