Representation rings for fusion systems and dimension functions
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.
We thank Matthew Gelvin, Jesper Grodal, and Bob Oliver for many helpful comments on the first version of the paper. Most of the work in this paper was carried out in May–June 2015 when the authors were visiting McMaster University. Both authors thank Ian Hambleton for hosting them at McMaster University. The second author would like to thank McMaster University for the support provided by a H. L. Hooker Visiting Fellowship.
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