Representation rings for fusion systems and dimension functions

TL;DR

This paper introduces a new framework for understanding representation rings of fusion systems, analyzing dimension functions of stable representations, and applying these concepts to finite group actions on homotopy spheres.

Contribution

It defines the representation ring for saturated fusion systems, studies dimension functions via transfer maps, and characterizes which super class functions are realizable as dimension functions.

Findings

Established conditions for dimension functions of $$-stable representations.

Connected dimension functions to realizability of super class functions.

Applied results to construct finite group actions on homotopy spheres.

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.