Skip to main content
SpringerLink
Log in

Representation rings for fusion systems and dimension functions

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bartel, A., Dokchitser, T.: Rational representations and permutation representations of finite groups. Math. Ann. 364(1–2), 539–558 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bauer, S.: A linearity theorem for group actions on spheres with applications to homotopy representations. Comment. Math. Helv. 64(1), 167–172 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bouc, S.: Biset Functors for Finite Groups, Lecture Notes in Mathematics, vol. 1990. Springer-Verlag, Berlin (2010)

    Book  MATH  Google Scholar 

  4. Bouc, S., Yalçın, E.: Borel–smith functions and the dade group. J. Algebra 311(2), 821–839 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. tom Dieck, T.: Transformation groups, de Gruyter Studies in Mathematics, vol. 8. Walter de Gruyter & Co., Berlin (1987)

    Book  Google Scholar 

  6. Dotzel, R.M., Hamrick, G.C.: \(p\)-group actions on homology spheres. Invent. Math. 62(3), 437–442 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dovermann, K.H., Petrie, T.: Artin relation for smooth representations. Proc. Natl. Acad. Sci. USA 77(10), 5620–5621 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gelvin, M., Reeh, S.P.: Minimal characteristic bisets for fusion systems. J. Algebra 427, 345–374 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Grodal J., Smith J.H.: Algebraic models for finite \(G\)-spaces. Oberwolfach Report 45, pp. 2690–2692 (2007)

  10. Hambleton, I., Yalçın, E.: Homotopy representations over the orbit category. Homol. Homotopy Appl. 16(2), 345–369 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hambleton, I., Yalçın, E.: Group actions on spheres with rank one prime power isotropy p 16, to appear in Math. Res. Lett., available at arXiv:1503.06298

  12. Jackson, M.A.: \({\text{ Qd }}(p)\)-free rank two finite groups act freely on a homotopy product of two spheres. J. Pure Appl. Algebra 208(3), 821–831 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Park, S.: Realizing a fusion system by a single finite group. Arch. Math. (Basel) 94(5), 405–410 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ragnarsson, K.: Classifying spectra of saturated fusion systems. Algebraic Geom. Topol. 6, 195–252 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ragnarsson, K., Stancu, R.: Saturated fusion systems as idempotents in the double burnside ring. Geom. Topol. 17(2), 839–904 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Reeh, S.P.: Transfer and characteristic idempotents for saturated fusion systems. Adv. Math. 289, 161–211 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Serre, J.-P.: Linear representations of finite groups. Springer-Verlag, New York-Heidelberg, Translated from the second French edition by Leonard L. Scott; Graduate Texts in Mathematics, Vol. 42 (1997)

  18. Suzuki, M.: Group theory. I, Grundlehren der Mathematischen Wissenschaften , vol 247. Springer-Verlag, Berlin-New York, (1982) Translated from the Japanese by the author

  19. Swan, R.G.: Periodic resolutions for finite groups. Ann. Math. (2) 72, 267–291 (1960)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Sune Precht Reeh

  2. Department of Mathematics, Bilkent University, 06800, Bilkent, Ankara, Turkey

    Ergün Yalçın

Authors
  1. Sune Precht Reeh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ergün Yalçın
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sune Precht Reeh.

Additional information

The first author is supported by the Danish Council for Independent Research’s Sapere Aude program (DFF–4002-00224). The second author is partially supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK) through the research program BİDEB-2219.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Reeh, S.P., Yalçın, E. Representation rings for fusion systems and dimension functions. Math. Z. 288, 509–530 (2018). https://doi.org/10.1007/s00209-017-1898-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-017-1898-8

Search

Navigation