The inductive McKay–Navarro conditions for the prime 2 and some groups of Lie type
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- by L. Ruhstorfer and A. A. Schaeffer Fry HTML | PDF
- Proc. Amer. Math. Soc. Ser. B 9 (2022), 204-220
Abstract:
For a prime $\ell$, the McKay conjecture suggests a bijection between the set of irreducible characters of a finite group with $\ell ’$-degree and the corresponding set for the normalizer of a Sylow $\ell$-subgroup. Navarro’s refinement suggests that the values of the characters on either side of this bijection should also be related, proposing that the bijection commutes with certain Galois automorphisms. Recently, Navarro–Späth–Vallejo have reduced the McKay–Navarro conjecture to certain “inductive” conditions on finite simple groups. We prove that these inductive McKay–Navarro (also called the inductive Galois–McKay) conditions hold for the prime $\ell =2$ for several groups of Lie type, namely the untwisted groups without non-trivial graph automorphisms.References
- Marc Cabanes and Michel Enguehard, Representation theory of finite reductive groups, New Mathematical Monographs, vol. 1, Cambridge University Press, Cambridge, 2004. MR 2057756, DOI 10.1017/CBO9780511542763
- François Digne and Jean Michel, Representations of finite groups of Lie type, London Mathematical Society Student Texts, vol. 21, Cambridge University Press, Cambridge, 1991. MR 1118841, DOI 10.1017/CBO9781139172417
- François Digne and Jean Michel, Groupes réductifs non connexes, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 3, 345–406 (French, with English and French summaries). MR 1272294, DOI 10.24033/asens.1696
- Meinolf Geck, Character values, Schur indices and character sheaves, Represent. Theory 7 (2003), 19–55. MR 1973366, DOI 10.1090/S1088-4165-03-00170-5
- Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups. Number 2. Part I. Chapter G, Mathematical Surveys and Monographs, vol. 40, American Mathematical Society, Providence, RI, 1996. General group theory. MR 1358135, DOI 10.1090/surv/040.2
- I. M. Isaacs, Gunter Malle, and Gabriel Navarro, A reduction theorem for the McKay conjecture, Invent. Math. 170 (2007), no. 1, 33–101. MR 2336079, DOI 10.1007/s00222-007-0057-y
- I. Martin Isaacs, Character theory of finite groups, AMS Chelsea Publishing, Providence, RI, 2006. Corrected reprint of the 1976 original [Academic Press, New York; MR0460423]. MR 2270898, DOI 10.1090/chel/359
- Birte Johansson, On the inductive Galois–McKay condition for finite groups of Lie type in their defining characteristic, Preprint, arXiv:2010.14837, 2020.
- Gunter Malle, Darstellungstheoretische Methoden bei der Realisierung einfacher Gruppen vom Lie Typ als Galoisgruppen, Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991) Progr. Math., vol. 95, Birkhäuser, Basel, 1991, pp. 443–459 (German). MR 1112174, DOI 10.1007/978-3-0348-8658-1_{2}0
- Gunter Malle, Height 0 characters of finite groups of Lie type, Represent. Theory 11 (2007), 192–220. MR 2365640, DOI 10.1090/S1088-4165-07-00312-3
- Gunter Malle, Extensions of unipotent characters and the inductive McKay condition, J. Algebra 320 (2008), no. 7, 2963–2980. MR 2442005, DOI 10.1016/j.jalgebra.2008.06.033
- Gunter Malle and Britta Späth, Characters of odd degree, Ann. of Math. (2) 184 (2016), no. 3, 869–908. MR 3549625, DOI 10.4007/annals.2016.184.3.6
- Gabriel Navarro, The McKay conjecture and Galois automorphisms, Ann. of Math. (2) 160 (2004), no. 3, 1129–1140. MR 2144975, DOI 10.4007/annals.2004.160.1129
- Gabriel Navarro, Britta Späth, and Carolina Vallejo, A reduction theorem for the Galois-McKay conjecture, Trans. Amer. Math. Soc. 373 (2020), no. 9, 6157–6183. MR 4155175, DOI 10.1090/tran/8111
- Lucas Ruhstorfer, The Navarro refinement of the McKay conjecture for finite groups of Lie type in defining characteristic, J. Algebra 582 (2021), 177–205. MR 4259262, DOI 10.1016/j.jalgebra.2021.04.025
- A. A. Schaeffer Fry, Galois automorphisms on Harish-Chandra series and Navarro’s self-normalizing Sylow 2-subgroup conjecture, Trans. Amer. Math. Soc. 372 (2019), no. 1, 457–483. MR 3968776, DOI 10.1090/tran/7590
- A. A. Schaeffer Fry, Galois-equivariant McKay bijections for primes dividing $q-1$, Israel J. Math., 2021, Online first, https://doi.org/10.1007/s11856-021-2266-2.
- A. A. Schaeffer Fry and Jay Taylor, Galois automorphisms and classical groups, Preprint, arXiv:2005.14088, 2020.
- Britta Späth, Inductive McKay condition in defining characteristic, Bull. Lond. Math. Soc. 44 (2012), no. 3, 426–438. MR 2966987, DOI 10.1112/blms/bdr100
Additional Information
- L. Ruhstorfer
- Affiliation: Fachbereich Mathematik, TU Kaiserslautern, 67653 Kaiserslautern, Germany
- MR Author ID: 1379611
- Email: ruhstorf@mathematik.uni-kl.de
- A. A. Schaeffer Fry
- Affiliation: Deptartment of Mathematics and Statistics, Metropolitan State University of Denver, Denver, Colorado 80217
- MR Author ID: 899206
- ORCID: 0000-0002-1690-9046
- Email: aschaef6@msudenver.edu
- Received by editor(s): June 28, 2021
- Received by editor(s) in revised form: January 24, 2022, and January 25, 2022
- Published electronically: April 25, 2022
- Additional Notes: The authors were supported by the Isaac Newton Institute for Mathematical Sciences in Cambridge and the organizers of the Spring 2020 INI program Groups, Representations, and Applications: New Perspectives, EPSRC grant EP/R014604/1, where this work began.
The second author was supported by the National Science Foundation (Award Nos. DMS-1801156 and DMS-2100912). - Communicated by: Martin Liebeck
- © Copyright 2022 by the authors under Creative Commons Attribution-NonCommercial 3.0 License (CC BY NC 3.0)
- Journal: Proc. Amer. Math. Soc. Ser. B 9 (2022), 204-220
- MSC (2000): Primary 20C15, 20C33
- DOI: https://doi.org/10.1090/bproc/123
- MathSciNet review: 4412275