# Representation rings for fusion systems and dimension functions

Representation rings for fusion systems and dimension functions We define the representation ring of a saturated fusion system $$\mathcal F$$ F as the Grothendieck ring of the semiring of $$\mathcal F$$ F -stable representations, and study the dimension functions of $$\mathcal F$$ F -stable representations using the transfer map induced by the characteristic idempotent of $$\mathcal F$$ F . We find a list of conditions for an $$\mathcal F$$ F -stable super class function to be realized as the dimension function of an $$\mathcal F$$ F -stable virtual representation. We also give an application of our results to constructions of finite group actions on homotopy spheres. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

# Representation rings for fusion systems and dimension functions

, Volume 288 (2) – May 10, 2017
22 pages

/lp/springer_journal/representation-rings-for-fusion-systems-and-dimension-functions-lByAg7xJOE
Publisher
Springer Journals
Copyright © 2017 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Mathematics, general
ISSN
0025-5874
eISSN
1432-1823
D.O.I.
10.1007/s00209-017-1898-8
Publisher site
See Article on Publisher Site

### Abstract

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

### Journal

Mathematische ZeitschriftSpringer Journals

Published: May 10, 2017

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