Complete reducibility in good characteristic
HTML articles powered by AMS MathViewer
- by Alastair J. Litterick and Adam R. Thomas PDF
- Trans. Amer. Math. Soc. 370 (2018), 5279-5340 Request permission
Abstract:
Let $G$ be a simple algebraic group of exceptional type, over an algebraically closed field of characteristic $p \ge 0$. A closed subgroup $H$ of $G$ is called $G$-completely reducible ($G$-cr) if whenever $H$ is contained in a parabolic subgroup $P$ of $G$, it is contained in a Levi subgroup of $P$. In this paper we determine the $G$-conjugacy classes of non-$G$-cr simple connected subgroups of $G$ when $p$ is good for $G$. For each such subgroup $X$, we determine the action of $X$ on the adjoint module $L(G)$ and the connected centraliser of $X$ in $G$. As a consequence we classify all non-$G$-cr connected reductive subgroups of $G$, and determine their connected centralisers. We also classify the subgroups of $G$ which are maximal among connected reductive subgroups, but not maximal among all connected subgroups.References
- Henning Haahr Andersen, Jens Jørgensen, and Peter Landrock, The projective indecomposable modules of $\textrm {SL}(2,\,p^{n})$, Proc. London Math. Soc. (3) 46 (1983), no. 1, 38–52. MR 684821, DOI 10.1112/plms/s3-46.1.38
- H. Azad, M. Barry, and G. Seitz, On the structure of parabolic subgroups, Comm. Algebra 18 (1990), no. 2, 551–562. MR 1047327, DOI 10.1080/00927879008823931
- Michael Bate, Benjamin Martin, and Gerhard Röhrle, A geometric approach to complete reducibility, Invent. Math. 161 (2005), no. 1, 177–218. MR 2178661, DOI 10.1007/s00222-004-0425-9
- Michael Bate, Benjamin Martin, and Gerhard Röhrle, Complete reducibility and commuting subgroups, J. Reine Angew. Math. 621 (2008), 213–235. MR 2431255, DOI 10.1515/CRELLE.2008.063
- Michael Bate, Benjamin Martin, Gerhard Röhrle, and Rudolf Tange, Complete reducibility and separability, Trans. Amer. Math. Soc. 362 (2010), no. 8, 4283–4311. MR 2608407, DOI 10.1090/S0002-9947-10-04901-9
- A. Borel and J. Tits, Éléments unipotents et sous-groupes paraboliques de groupes réductifs. I, Invent. Math. 12 (1971), 95–104 (French). MR 294349, DOI 10.1007/BF01404653
- Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
- N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
- Roger W. Carter, Simple groups of Lie type, Pure and Applied Mathematics, Vol. 28, John Wiley & Sons, London-New York-Sydney, 1972. MR 0407163
- Peter D. Lax, Linear algebra, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1997. A Wiley-Interscience Publication. MR 1423602
- E. Cline, B. Parshall, L. Scott, and Wilberd van der Kallen, Rational and generic cohomology, Invent. Math. 39 (1977), no. 2, 143–163. MR 439856, DOI 10.1007/BF01390106
- Stephen Donkin, A filtration for rational modules, Math. Z. 177 (1981), no. 1, 1–8. MR 611465, DOI 10.1007/BF01214334
- David S. Dummit and Richard M. Foote, Abstract algebra, 3rd ed., John Wiley & Sons, Inc., Hoboken, NJ, 2004. MR 2286236
- Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups. Number 3. Part I. Chapter A, Mathematical Surveys and Monographs, vol. 40, American Mathematical Society, Providence, RI, 1998. Almost simple $K$-groups. MR 1490581, DOI 10.1090/surv/040.3
- Jens Carsten Jantzen, Low-dimensional representations of reductive groups are semisimple, Algebraic groups and Lie groups, Austral. Math. Soc. Lect. Ser., vol. 9, Cambridge Univ. Press, Cambridge, 1997, pp. 255–266. MR 1635685
- Jens Carsten Jantzen, Representations of algebraic groups, 2nd ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR 2015057
- Peter Kleidman and Martin Liebeck, The subgroup structure of the finite classical groups, London Mathematical Society Lecture Note Series, vol. 129, Cambridge University Press, Cambridge, 1990. MR 1057341, DOI 10.1017/CBO9780511629235
- Martin W. Liebeck and Gary M. Seitz, Reductive subgroups of exceptional algebraic groups, Mem. Amer. Math. Soc. 121 (1996), no. 580, vi+111. MR 1329942, DOI 10.1090/memo/0580
- Paul Fong, The Isaacs-Navarro conjecture for symmetric groups, J. Algebra 260 (2003), no. 1, 154–161. Special issue celebrating the 80th birthday of Robert Steinberg. MR 1973581, DOI 10.1016/S0021-8693(02)00630-0
- Martin W. Liebeck and Gary M. Seitz, The maximal subgroups of positive dimension in exceptional algebraic groups, Mem. Amer. Math. Soc. 169 (2004), no. 802, vi+227. MR 2044850, DOI 10.1090/memo/0802
- Martin W. Liebeck and Gary M. Seitz, Unipotent and nilpotent classes in simple algebraic groups and Lie algebras, Mathematical Surveys and Monographs, vol. 180, American Mathematical Society, Providence, RI, 2012. MR 2883501, DOI 10.1090/surv/180
- Martin W. Liebeck and Donna M. Testerman, Irreducible subgroups of algebraic groups, Q. J. Math. 55 (2004), no. 1, 47–55. MR 2043006, DOI 10.1093/qjmath/55.1.47
- Frank Lübeck, Small degree representations of finite Chevalley groups in defining characteristic, LMS J. Comput. Math. 4 (2001), 135–169. MR 1901354, DOI 10.1112/S1461157000000838
- George J. McNinch, Dimensional criteria for semisimplicity of representations, Proc. London Math. Soc. (3) 76 (1998), no. 1, 95–149. MR 1476899, DOI 10.1112/S0024611598000045
- George J. McNinch, Levi decompositions of a linear algebraic group, Transform. Groups 15 (2010), no. 4, 937–964. MR 2753264, DOI 10.1007/s00031-010-9111-8
- Gary M. Seitz, Maximal subgroups of exceptional algebraic groups, Mem. Amer. Math. Soc. 90 (1991), no. 441, iv+197. MR 1048074, DOI 10.1090/memo/0441
- Gary M. Seitz, Unipotent elements, tilting modules, and saturation, Invent. Math. 141 (2000), no. 3, 467–502. MR 1779618, DOI 10.1007/s002220000073
- Jean-Pierre Serre, Morsund lectures, University of Oregon, 1998.
- Jean-Pierre Serre, Galois cohomology, Springer-Verlag, Berlin, 1997. Translated from the French by Patrick Ion and revised by the author. MR 1466966, DOI 10.1007/978-3-642-59141-9
- Jean-Pierre Serre, Complète réductibilité, Astérisque 299 (2005), Exp. No. 932, viii, 195–217 (French, with French summary). Séminaire Bourbaki. Vol. 2003/2004. MR 2167207
- Robert Steinberg, Lectures on Chevalley groups, Yale University, New Haven, Conn., 1968. Notes prepared by John Faulkner and Robert Wilson. MR 0466335
- David I. Stewart, The second cohomology of simple $\textrm {SL}_2$-modules, Proc. Amer. Math. Soc. 138 (2010), no. 2, 427–434. MR 2557160, DOI 10.1090/S0002-9939-09-10088-6
- David I. Stewart, Non-G-completely reducible subgroups of the exceptional algebraic groups, International Mathematics Research Notices (2013).
- David I. Stewart, On unipotent algebraic $G$-groups and 1-cohomology, Trans. Amer. Math. Soc. 365 (2013), no. 12, 6343–6365. MR 3105754, DOI 10.1090/S0002-9947-2013-05853-9
- David I. Stewart, The reductive subgroups of $F_4$, Mem. Amer. Math. Soc. 223 (2013), no. 1049, vi+88. MR 3075783, DOI 10.1090/S0065-9266-2012-00668-X
- Adam R. Thomas, Simple irreducible subgroups of exceptional algebraic groups, J. Algebra 423 (2015), 190–238. MR 3283715, DOI 10.1016/j.jalgebra.2014.10.011
- Adam R. Thomas, Irreducible $A_1$ subgroups of exceptional algebraic groups, J. Algebra 447 (2016), 240–296. MR 3427635, DOI 10.1016/j.jalgebra.2015.08.026
Additional Information
- Alastair J. Litterick
- Affiliation: Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany
- Address at time of publication: Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstraße 150, D-44780 Bochum, Germany
- MR Author ID: 924128
- Email: ajlitterick@gmail.com
- Adam R. Thomas
- Affiliation: Heilbronn Institute for Mathematical Research, University of Bristol, Bristol, United Kingdom
- Address at time of publication: School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK, and The Heilbronn Institute for Mathematical Research, Bristol, United Kingdom
- MR Author ID: 1091953
- Email: adamthomas22@gmail.com
- Received by editor(s): September 14, 2015
- Received by editor(s) in revised form: March 7, 2016, September 26, 2016, and September 30, 2016
- Published electronically: April 17, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 5279-5340
- MSC (2010): Primary 20G07, 20G41; Secondary 20G10
- DOI: https://doi.org/10.1090/tran/7085
- MathSciNet review: 3803140