Representation rings for fusion systems and dimension functions

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