A Murnaghan–Nakayama rule for values of unipotent characters in classical groups
HTML articles powered by AMS MathViewer
- by Frank Lübeck and Gunter Malle PDF
- Represent. Theory 20 (2016), 139-161 Request permission
Corrigendum: Represent. Theory 21 (2017), 1-3.
Abstract:
We derive a Murnaghan–Nakayama type formula for the values of unipotent characters of finite classical groups on regular semisimple elements. This relies on Asai’s explicit decomposition of Lusztig restriction. We use our formula to show that most complex irreducible characters vanish on some $\ell$-singular element for certain primes $\ell$.
As an application we classify the simple endotrivial modules of the finite quasi-simple classical groups. As a further application we show that for finite simple classical groups and primes $\ell \ge 3$ the first Cartan invariant in the principal $\ell$-block is larger than 2 unless Sylow $\ell$-subgroups are cyclic.
References
- Teruaki Asai, Unipotent class functions of split special orthogonal groups $\textrm {SO}^{+}_{2n}$ over finite fields, Comm. Algebra 12 (1984), no. 5-6, 517–615. MR 735137, DOI 10.1080/00927878408823017
- Teruaki Asai, The unipotent class functions on the symplectic groups and the odd orthogonal groups over finite fields, Comm. Algebra 12 (1984), no. 5-6, 617–645. MR 735138, DOI 10.1080/00927878408823018
- Teruaki Asai, The unipotent class functions of nonsplit finite special orthogonal groups, Comm. Algebra 13 (1985), no. 4, 845–924. MR 776867, DOI 10.1080/00927878508823197
- Roger W. Carter, Finite groups of Lie type, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR 794307
- 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
- Paul Fong and Bhama Srinivasan, Generalized Harish-Chandra theory for unipotent characters of finite classical groups, J. Algebra 104 (1986), no. 2, 301–309. MR 866777, DOI 10.1016/0021-8693(86)90217-6
- Meinolf Geck and Nicolas Jacon, Representations of Hecke algebras at roots of unity, Algebra and Applications, vol. 15, Springer-Verlag London, Ltd., London, 2011. MR 2799052, DOI 10.1007/978-0-85729-716-7
- Gordon James and Andrew Mathas, The Jantzen sum formula for cyclotomic $q$-Schur algebras, Trans. Amer. Math. Soc. 352 (2000), no. 11, 5381–5404. MR 1665333, DOI 10.1090/S0002-9947-00-02492-2
- Shigeo Koshitani, Burkhard Külshammer, and Benjamin Sambale, On Loewy lengths of blocks, Math. Proc. Cambridge Philos. Soc. 156 (2014), no. 3, 555–570. MR 3181640, DOI 10.1017/S0305004114000103
- Caroline Lassueur and Gunter Malle, Simple endotrivial modules for linear, unitary and exceptional groups, Math. Z. 280 (2015), no. 3-4, 1047–1074. MR 3369366, DOI 10.1007/s00209-015-1465-0
- C. Lassueur, G. Malle, E. Schulte, Simple endotrivial modules for quasi-simple groups, J. Reine Angew. Math. DOI: 10.1515/crelle-2013-0100.
- George Lusztig, Characters of reductive groups over a finite field, Annals of Mathematics Studies, vol. 107, Princeton University Press, Princeton, NJ, 1984. MR 742472, DOI 10.1515/9781400881772
- Gunter Malle and Donna Testerman, Linear algebraic groups and finite groups of Lie type, Cambridge Studies in Advanced Mathematics, vol. 133, Cambridge University Press, Cambridge, 2011. MR 2850737, DOI 10.1017/CBO9780511994777
- Christopher Parker and Peter Rowley, A characteristic 5 identification of the Lyons group, J. London Math. Soc. (2) 69 (2004), no. 1, 128–140. MR 2025331, DOI 10.1112/S0024610703004848
- Götz Pfeiffer, Character tables of Weyl groups in GAP, Bayreuth. Math. Schr. 47 (1994), 165–222. MR 1285208
Additional Information
- Frank Lübeck
- Affiliation: Lehrstuhl D für Mathematik, RWTH Aachen, Pontdriesch 14/16, 52062 Aachen, Germany.
- MR Author ID: 362381
- Email: Frank.Luebeck@math.rwth-aachen.de
- Gunter Malle
- Affiliation: FB Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany.
- MR Author ID: 225462
- Email: malle@mathematik.uni-kl.de
- Received by editor(s): August 11, 2015
- Received by editor(s) in revised form: January 11, 2016
- Published electronically: March 4, 2016
- Additional Notes: The second author gratefully acknowledges financial support by ERC Advanced Grant 291512.
- © Copyright 2016 American Mathematical Society
- Journal: Represent. Theory 20 (2016), 139-161
- MSC (2010): Primary 20C20; Secondary 20C33, 20C34
- DOI: https://doi.org/10.1090/ert/480
- MathSciNet review: 3466537