Skip to main content

Dade’s ordinary conjecture implies the Alperin–McKay conjecture

Abstract

We show that Dade’s ordinary conjecture implies the Alperin–McKay conjecture. We remark that some of the methods can be used to identify a canonical height zero character in a nilpotent block.

References

  1. Broué, M., Puig, L.: Characters and local structure in \(G\)-algebras. J. Algebra 63, 306–317 (1980)

    MathSciNet  Article  MATH  Google Scholar 

  2. Dade, E.C.: A correspondence of characters, The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), In: Proceedings of Symposia in Pure Mathematics, vol. 37, pp. 401–403. American Mathematical Society, Providence (1980)

  3. Dade, E.C.: Counting characters in blocks, I. Invent. Math. 109, 187–210 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  4. Dade, E.C.: Counting characters in blocks, II. J. Reine Angew. Math. 448, 97–190 (1994)

    MathSciNet  MATH  Google Scholar 

  5. Diaz, A., Glesser, A., Mazza, N., Park, S.: Control of transfer and weak closure in fusion systems. J. Algebra 323, 382–392 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  6. Linckelmann, M.: Alperin’s weight conjecture in terms of equivariant Bredon cohomology. Math. Z. 250, 495–513 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  7. Linckelmann, M.: The orbit space of a fusion system is contractible. Proc. Lond. Math. Soc. 98, 191–216 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  8. Linckelmann, M.: On automorphisms and focal subgroups of blocks. Preprint (2016). arXiv:1612.07739

  9. Murai, M.: Block induction, normal subgroups and characters of height zero. Osaka J. Math. 31, 9–25 (1994)

    MathSciNet  MATH  Google Scholar 

  10. Murai, M.: On a minimal counterexample to the Alperin–McKay conjecture. Proc. Japan Acad. 87, 192–193 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  11. Navarro, G.: Character theory and the McKay conjecture. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2018). https://doi.org/10.1017/9781108552790

  12. Okuyama, T., Wajima, M.: Character correspondence and \(p\)-blocks of \(p\)-solvable groups. Osaka J. Math. 17, 801–806 (1980)

    MathSciNet  MATH  Google Scholar 

  13. Picaronny, C., Puig, L.: Quelques remarques sur un thème de Knörr. J. Algebra 109, 69–73 (1987)

    MathSciNet  Article  MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

  15. Robinson, G.R.: Weight conjectures for ordinary characters. J. Algebra 276, 761–775 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  16. Robinson, G.R.: On the focal defect group of a block, characters of height zero, and lower defect group multiplicities. J. Algebra 320(6), 2624–2628 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  17. Sambale, B.: On the projective height zero conjecture. arXiv:1801.04272

Download references

Acknowledgements

We thank Gunter Malle for helpful comments on the first version of the paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Radha Kessar.

Additional information

This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2018 semester. The second author acknowledges support from EPSRC Grant EP/M02525X/1.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kessar, R., Linckelmann, M. Dade’s ordinary conjecture implies the Alperin–McKay conjecture. Arch. Math. 112, 19–25 (2019). https://doi.org/10.1007/s00013-018-1230-9

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00013-018-1230-9

Keywords

  • Dade’s ordinary conjecture
  • Alperin–McKay conjecture
  • Height zero characters

Mathematics Subject Classification

  • 20C20