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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Broué, M., Puig, L.: Characters and local structure in \(G\)-algebras. J. Algebra 63, 306–317 (1980)
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)
Dade, E.C.: Counting characters in blocks, I. Invent. Math. 109, 187–210 (1992)
Dade, E.C.: Counting characters in blocks, II. J. Reine Angew. Math. 448, 97–190 (1994)
Diaz, A., Glesser, A., Mazza, N., Park, S.: Control of transfer and weak closure in fusion systems. J. Algebra 323, 382–392 (2010)
Linckelmann, M.: Alperin’s weight conjecture in terms of equivariant Bredon cohomology. Math. Z. 250, 495–513 (2005)
Linckelmann, M.: The orbit space of a fusion system is contractible. Proc. Lond. Math. Soc. 98, 191–216 (2009)
Linckelmann, M.: On automorphisms and focal subgroups of blocks. Preprint (2016). arXiv:1612.07739
Murai, M.: Block induction, normal subgroups and characters of height zero. Osaka J. Math. 31, 9–25 (1994)
Murai, M.: On a minimal counterexample to the Alperin–McKay conjecture. Proc. Japan Acad. 87, 192–193 (2011)
Navarro, G.: Character theory and the McKay conjecture. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2018). https://doi.org/10.1017/9781108552790
Okuyama, T., Wajima, M.: Character correspondence and \(p\)-blocks of \(p\)-solvable groups. Osaka J. Math. 17, 801–806 (1980)
Picaronny, C., Puig, L.: Quelques remarques sur un thème de Knörr. J. Algebra 109, 69–73 (1987)
Puig, L.: Nilpotent blocks and their source algebras. Invent. Math. 93, 77–116 (1988)
Robinson, G.R.: Weight conjectures for ordinary characters. J. Algebra 276, 761–775 (2004)
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)
Sambale, B.: On the projective height zero conjecture. arXiv:1801.04272
Acknowledgements
We thank Gunter Malle for helpful comments on the first version of the paper.
Author information
Authors and Affiliations
Corresponding author
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.
About this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-018-1230-9