A reduction theorem for the Galois–McKay conjecture

Navarro, Gabriel; Späth, Britta; Vallejo, Carolina

doi:10.1090/tran/8111

A reduction theorem for the Galois-McKay conjecture

Authors: Gabriel Navarro, Britta Späth and Carolina Vallejo
Journal: Trans. Amer. Math. Soc. 373 (2020), 6157-6183
MSC (2010): Primary 20C15; Secondary 20C25
DOI: https://doi.org/10.1090/tran/8111
Published electronically: June 24, 2020
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Abstract: We introduce ${\mathcal {H}}$ -triples and a partial order relation on them, generalizing the theory of ordering character triples developed by Navarro and Späth. This generalization takes into account the action of Galois automorphisms on characters and, together with previous results of Ladisch and Turull, allows us to reduce the Galois-McKay conjecture to a question about simple groups.

[Bra45] Richard Brauer, On the representation of a group of order 𝑔 in the field of the 𝑔-th roots of unity, Amer. J. Math. 67 (1945), 461–471. MR 14085, https://doi.org/10.2307/2371973
[BN18] Olivier Brunat, Rishi Nath.
The Navarro Conjecture for the alternating groups.
ArXiv:1803.01423.
[GAP] The GAP Group, GAP Groups, Algorithms, and Programming, Version 4.4; 2004, http://www.gap-system.org.
[Isa06] 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
[IMN07] 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, https://doi.org/10.1007/s00222-007-0057-y
[Lad16] Frieder Ladisch, On Clifford theory with Galois action, J. Algebra 457 (2016), 45–72. MR 3490077, https://doi.org/10.1016/j.jalgebra.2016.03.008
[Mal19] Gunter Malle, The Navarro-Tiep Galois conjecture for 𝑝=2, Arch. Math. (Basel) 112 (2019), no. 5, 449–457. MR 3943465, https://doi.org/10.1007/s00013-019-01298-6
[MS16] Gunter Malle and Britta Späth, Characters of odd degree, Ann. of Math. (2) 184 (2016), no. 3, 869–908. MR 3549625, https://doi.org/10.4007/annals.2016.184.3.6
[McK72] John McKay, Irreducible representations of odd degree, J. Algebra 20 (1972), 416–418. MR 286904, https://doi.org/10.1016/0021-8693(72)90066-X
[Nat09] Rishi Nath, The Navarro conjecture for the alternating groups, 𝑝=2, J. Algebra Appl. 8 (2009), no. 6, 837–844. MR 2597284, https://doi.org/10.1142/S0219498809003667
[Nav04] Gabriel Navarro, The McKay conjecture and Galois automorphisms, Ann. of Math. (2) 160 (2004), no. 3, 1129–1140. MR 2144975, https://doi.org/10.4007/annals.2004.160.1129
[Nav18] Gabriel Navarro, Character theory and the McKay conjecture, Cambridge Studies in Advanced Mathematics, vol. 175, Cambridge University Press, Cambridge, 2018. MR 3753712
[NS14] Gabriel Navarro and Britta Späth, On Brauer’s height zero conjecture, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 4, 695–747. MR 3191974, https://doi.org/10.4171/JEMS/444
[NT19] Gabriel Navarro and Pham Huu Tiep, Sylow subgroups, exponents, and character values, Trans. Amer. Math. Soc. 372 (2019), no. 6, 4263–4291. MR 4009430, https://doi.org/10.1090/tran/7816
[NTT07] Gabriel Navarro, Pham Huu Tiep, and Alexandre Turull, 𝑝-rational characters and self-normalizing Sylow 𝑝-subgroups, Represent. Theory 11 (2007), 84–94. MR 2306612, https://doi.org/10.1090/S1088-4165-07-00263-4
[Rey65] W. F. Reynolds, Projective representations of finite groups in cyclotomic fields, Illinois J. Math. 9 (1965), 191–198. MR 174631
[Ruh17] Lucas Ruhstorfer,
The Navarro refinement of the McKay conjecture for finite groups of Lie type in defining characteristic,
ArXiv:1703.09006.
[SF18] 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, https://doi.org/10.1090/tran/7590
[Spä14] Britta Späth, A reduction theorem for Dade’s projective conjecture, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 4, 1071–1126. MR 3626551, https://doi.org/10.4171/JEMS/688
[Spä20] Britta Späth,
The inductive Galois-McKay condition for ${\mathrm {PSL}}_2(q)$ , in preparation.
[Tur08a] Alexandre Turull, Strengthening the McKay conjecture to include local fields and local Schur indices, J. Algebra 319 (2008), no. 12, 4853–4868. MR 2423808, https://doi.org/10.1016/j.jalgebra.2005.12.035
[Tur08b] Alexandre Turull, Above the Glauberman correspondence, Adv. Math. 217 (2008), no. 5, 2170–2205. MR 2388091, https://doi.org/10.1016/j.aim.2007.10.001
[Tur09] Alexandre Turull, The Brauer-Clifford group, J. Algebra 321 (2009), no. 12, 3620–3642. MR 2517805, https://doi.org/10.1016/j.jalgebra.2009.02.019
[Tur13] Alexandre Turull, The strengthened Alperin-McKay conjecture for 𝑝-solvable groups, J. Algebra 394 (2013), 79–91. MR 3092712, https://doi.org/10.1016/j.jalgebra.2013.06.028
[Tur17] Alexandre Turull, Endoisomorphisms and character triple isomorphisms, J. Algebra 474 (2017), 466–504. MR 3595799, https://doi.org/10.1016/j.jalgebra.2016.10.048

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Additional Information

Gabriel Navarro
Affiliation: Departament de Matemàtiques, Universitat de València, 46100 Burjassot, València, Spain
MR Author ID: 129760
Email: gabriel.navarro@uv.es

Britta Späth
Affiliation: BU Wuppertal, Gaußstrasse 20, 42119 Wuppertal, Germany
Email: bspaeth@uni-wuppertal.de

Carolina Vallejo
Affiliation: Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049 Madrid, Spain
MR Author ID: 1001337
Email: carolina.vallejo@uam.es

DOI: https://doi.org/10.1090/tran/8111
Keywords: Galois action on characters, Galois--McKay conjecture, reduction theorem
Received by editor(s): June 27, 2019
Received by editor(s) in revised form: December 13, 2019
Published electronically: June 24, 2020
Additional Notes: This material is partially 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 research of the first and third-named authors was partially supported by Ministerio de Cienciae Innovación PID2019-103854GB-I00 and FEDER Funds. The research of the second-named author was supported by the research training group GRK 2240: Algebro-Geometric Methods in Algebra, Arithmetic and Topology, funded by the DFG. The third-named author also acknowledges support by the ICMAT Severo Ochoa project SEV-2011-0087.
Article copyright: © Copyright 2020 American Mathematical Society