Skip to main content
SpringerLink
Log in

Descent of Equivalences and Character Bijections

  • Conference paper
  • First Online:
Geometric and Topological Aspects of the Representation Theory of Finite Groups (PSSW 2016)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 242))

Included in the following conference series:

Abstract

Categorical equivalences between block algebras of finite groups—such as Morita and derived equivalences—are well known to induce character bijections which commute with the Galois groups of field extensions. This is the motivation for attempting to realise known Morita and derived equivalences over non-splitting fields. This article presents various results on the theme of descent to appropriate subfields and subrings. We start with the observation that perfect isometries induced by a virtual Morita equivalence induce isomorphisms of centres in non-split situations and explain connections with Navarro’s generalisation of the Alperin–McKay conjecture. We show that Rouquier’s splendid Rickard complex for blocks with cyclic defect groups descends to the non-split case. We also prove a descent theorem for Morita equivalences with endopermutation source.

Part of the work on this paper was done while the authors were visiting the University of California, Santa Cruz, and they would like to thank the Mathematics Department for its hospitality. The second author acknowledges support from EPSRC grant EP/M02525X/1.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. R. Boltje and B. Xu, On p-permutation equivalences: between Rickard equivalences and isotypies, Trans. Amer. Math. Soc 360 (2008), 5067–5087.

    Article  MathSciNet  Google Scholar 

  2. S. Bouc, The Dade group of a p-group, Invent. Math., 164 (2006), 189–231.

    Article  MathSciNet  Google Scholar 

  3. M. Broué, Isométries parfaites, types de blocs, catégories dérivées, Astérisque 181–182 (1990), 61–92.

    MATH  Google Scholar 

  4. C. W. Curtis and I. Reiner, Methods of Representation theory Vol. I, John Wiley and Sons, New York, London, Sydney (1981).

    MATH  Google Scholar 

  5. E. C. Dade, Endo-permutation modules over p-groups, I, II, Ann. Math. 107 (1978), 459–494, 108 (1978), 317–346.

    Google Scholar 

  6. W. Feit, The Representation Theory of Finite Groups, North-Holland Mathematical Library 25, North-Holland Publishing Company, Amsterdam (1982).

    Google Scholar 

  7. J. A. Green, On the indecomposable representations of a finite group, Math. Z. 70 (1959), 430–445.

    Article  MathSciNet  Google Scholar 

  8. J. A. Green, Walking around the Brauer tree, J. Austr. Math. Soc. 17 (1974), 197–213.

    Article  MathSciNet  Google Scholar 

  9. A. Grothendieck, Éléments de géométrie algébrique IV, Étude locale des schémas et des morphismes de schémas II. Inst. Hautes Etudes Sci. Publ. Math. 24 (1965).

    Google Scholar 

  10. R. Knörr, Relative projective covers. Proc. Symp. Mod. Repr. Finite Groups, Aarhus Univ. (1978), 28–32.

    Google Scholar 

  11. B. Külshammer, On p-blocks of p-solvable groups, Comm. Algebra 9 (1981), 1763–1785.

    Article  MathSciNet  Google Scholar 

  12. B. Külshammer, Crossed products and blocks with normal defect groups, Comm. Algebra 13 (1985), 147–168.

    Article  MathSciNet  Google Scholar 

  13. T. Y. Lam, Lectures on Modules and Rings. Graduate Texts in Mathematics 189, Springer-Verlag, New York Berlin Heidelgerg (1998).

    Google Scholar 

  14. C. Lassueur and J. Thévenaz, Lifting endo-p-permutation modules. Preprint (2017).

    Google Scholar 

  15. M. Linckelmann, Modules in the sources of Green’s exact sequences for cyclic blocks, Invent. Math. 97 (1989) 129–140.

    Article  MathSciNet  Google Scholar 

  16. M. Linckelmann, Stable equivalences of Morita type for self-injective algebras and p-groups, Math. Z. 223 (1996) 87–100.

    Article  MathSciNet  Google Scholar 

  17. M. Linckelmann, The isomorphism problem for cyclic blocks and their source algebras, Invent. Math. 125, (1996), 265–283.

    Article  MathSciNet  Google Scholar 

  18. M. Linckelmann, Transfer in Hochschild cohomology of blocks of finite groups, Algebras and Representation Theory 2 (1999), 107–135.

    Article  MathSciNet  Google Scholar 

  19. M. Linckelmann, On stable equivalences with endopermutation source, J. Algebra 434 (2015), 27–45.

    Article  MathSciNet  Google Scholar 

  20. M. Linckelmann, The block theory of finite groups I/II. Cambridge University Press, to appear (2018).

    Google Scholar 

  21. M. Linckelmann and N. Mazza, The Dade group of a fusion system. J. Group Theory 12 (2009), 55–74.

    Article  MathSciNet  Google Scholar 

  22. S. Mac Lane, Categories for the working mathematician, Graduate Texts in Mathematics 5 (1998).

    Google Scholar 

  23. A. Mărcuş, Derived invariance of Cliord classes, J. Group Theory 12 (2009), 83–94.

    Article  MathSciNet  Google Scholar 

  24. H. Nagao and Y. Tsushima, Representations of Finite Groups, Academic Press, San Diego (1989).

    MATH  Google Scholar 

  25. G. Navarro, The McKay conjecture and Galois automorphisms, Annals of Mathematics, 160 (2004), 1129–1140.

    Article  MathSciNet  Google Scholar 

  26. J. Neukirch, Algebraic Number Theory, Springer-Verlag, Berlin (1999).

    Book  Google Scholar 

  27. L. Puig, Nilpotent blocks and their source algebras, Invent. Math. 93 (1988), 77–116.

    Article  MathSciNet  Google Scholar 

  28. L. Puig, On the local Structure of Morita and Rickard Equivalences Between Brauer blocks, Progr. Math., 178, Birkhäuser, Basel, 1999.

    Google Scholar 

  29. L. Puig, Block source algebras in p-solvable groups, Michigan Math. J. 58 (2009), 323–338.

    Article  MathSciNet  Google Scholar 

  30. I. Reiner, Relations between integral and modular representations, Michigan Math. J. 13 (1966) 357–372.

    Article  MathSciNet  Google Scholar 

  31. K. W. Roggenkamp, Blocks of cyclic defect and Green orders. Comm. Algebra 20(6) (1992), 1715–1734.

    Article  MathSciNet  Google Scholar 

  32. R. Rouquier, The derived category of blocks with cyclic defect groups, in: Derived Equivalences for Group Rings (S. König, A. Zimmermann), Lecture Notes in Math. 1685, Springer Verlag, Berlin-Heidelberg, 1998, 199–220.

    Google Scholar 

  33. J. P. Serre, Local fields, Graduate Texts in Mathematics 67 (1979).

    Google Scholar 

  34. J. Thévenaz, Relative projective covers and almost split sequences. Comm. Alg. 13 (1985), 1535–1554.

    Article  MathSciNet  Google Scholar 

  35. J. Thévenaz, G-Algebras and Modular Representation Theory, Oxford Science Publications, Clarendon, Oxford (1995).

    MATH  Google Scholar 

  36. J. Thévenaz, Endo-permutation Modules, a guided tour in: Group Representation Theory (M. Geck, D. Testerman, J. Thévenaz), EPFL Press, 2007, 115–147.

    Google Scholar 

  37. A. Turull, Strengthening the McKay conjecture to include local fields and local Schur indices, J. Algebra 319 (2008), 4853–4868.

    Article  MathSciNet  Google Scholar 

  38. C. A. Weibel, An introduction to homological algebra, Cambridge studies in advanced mathematics 38, Cambridge University Press (1994).

    Google Scholar 

  39. A. Zimmermann, Two Sided Tilting Complexes for Green Orders and Brauer Tree Algebras. J. Algebra 187 (1997), 446–473.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of London, London, EC1V 0HB, UK

    Radha Kessar & Markus Linckelmann

Authors
  1. Radha Kessar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Markus Linckelmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Radha Kessar .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Georgia, Athens, GA, USA

    Jon F. Carlson

  2. Department of Mathematics, University of Utah, Salt Lake City, UT, USA

    Srikanth B. Iyengar

  3. Department of Mathematics, University of Washington, Seattle, WA, USA

    Julia Pevtsova

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Kessar, R., Linckelmann, M. (2018). Descent of Equivalences and Character Bijections. In: Carlson, J., Iyengar, S., Pevtsova, J. (eds) Geometric and Topological Aspects of the Representation Theory of Finite Groups. PSSW 2016. Springer Proceedings in Mathematics & Statistics, vol 242. Springer, Cham. https://doi.org/10.1007/978-3-319-94033-5_7

Download citation

Publish with us

Policies and ethics

Search

Navigation