AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Irreducible almost simple subgroups of classical algebraic groups
About this Title
Timothy C. Burness, School of Mathematics, University of Bristol, Bristol, BS8 1TW, United Kingdom, Soumaïa Ghandour, Faculté des Sciences, Section V, Université Libanaise, Nabatieh, Lebanon, Claude Marion, Section de Mathématiques, Station 8, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland and Donna M. Testerman, Section de Mathématiques, Station 8, École Polytechnique Fédér-, ale de Lausanne, CH-1015 Lausanne, Switzerland
Publication: Memoirs of the American Mathematical Society
Publication Year:
2015; Volume 236, Number 1114
ISBNs: 978-1-4704-1046-9 (print); 978-1-4704-2280-6 (online)
DOI: https://doi.org/10.1090/memo/1114
Published electronically: December 16, 2014
Keywords: Classical algebraic group; disconnected maximal subgroup; irreducible triple
MSC: Primary 20G05; Secondary 20E28, 20E32
Table of Contents
Chapters
- 1. Introduction
- 2. Preliminaries
- 3. The case $H^0 = A_m$
- 4. The case $H^0=D_m$, $m \ge 5$
- 5. The case $H^0=E_6$
- 6. The case $H^0 = D_4$
- 7. Proof of Theorem
- Notation
Abstract
Let $G$ be a simple classical algebraic group over an algebraically closed field $K$ of characteristic $p\geq 0$ with natural module $W$. Let $H$ be a closed subgroup of $G$ and let $V$ be a nontrivial $p$-restricted irreducible tensor indecomposable rational $KG$-module such that the restriction of $V$ to $H$ is irreducible. In this paper we classify the triples $(G,H,V)$ of this form, where $V \neq W,W^{*}$ and $H$ is a disconnected almost simple positive-dimensional closed subgroup of $G$ acting irreducibly on $W$. Moreover, by combining this result with earlier work, we complete the classification of the irreducible triples $(G,H,V)$ where $G$ is a simple algebraic group over $K$, and $H$ is a maximal closed subgroup of positive dimension.- Michael Aschbacher and Gary M. Seitz, Involutions in Chevalley groups over fields of even order, Nagoya Math. J. 63 (1976), 1–91. MR 422401
- V. Balaji and János Kollár, Holonomy groups of stable vector bundles, Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 183–211. MR 2426347, DOI 10.2977/prims/1210167326
- Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 4–6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002. Translated from the 1968 French original by Andrew Pressley. MR 1890629
- Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 7–9, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2005. Translated from the 1975 and 1982 French originals by Andrew Pressley. MR 2109105
- Jonathan Brundan, Double coset density in classical algebraic groups, Trans. Amer. Math. Soc. 352 (2000), no. 3, 1405–1436. MR 1751310, DOI 10.1090/S0002-9947-99-02258-8
- T.C. Burness, S. Ghandour and D.M. Testerman, Irreducible geometric subgroups of classical algebraic groups, Mem. Amer. Math. Soc., to appear.
- E. Dynkin, Maximal subgroups of the classical groups, Amer. Math. Soc. Translations 6 (1957), 245–378.
- Ben Ford, Overgroups of irreducible linear groups. I, J. Algebra 181 (1996), no. 1, 26–69. MR 1382025, DOI 10.1006/jabr.1996.0108
- Ben Ford, Overgroups of irreducible linear groups. II, Trans. Amer. Math. Soc. 351 (1999), no. 10, 3869–3913. MR 1467464, DOI 10.1090/S0002-9947-99-02138-8
- Ben Ford and Alexander S. Kleshchev, A proof of the Mullineux conjecture, Math. Z. 226 (1997), no. 2, 267–308. MR 1477629, DOI 10.1007/PL00004340
- Soumaïa Ghandour, Irreducible disconnected subgroups of exceptional algebraic groups, J. Algebra 323 (2010), no. 10, 2671–2709. MR 2609170, DOI 10.1016/j.jalgebra.2010.02.018
- Robert M. Guralnick and Pham Huu Tiep, Decompositions of small tensor powers and Larsen’s conjecture, Represent. Theory 9 (2005), 138–208. MR 2123127, DOI 10.1090/S1088-4165-05-00192-5
- Robert M. Guralnick and Pham Huu Tiep, Symmetric powers and a problem of Kollár and Larsen, Invent. Math. 174 (2008), no. 3, 505–554. MR 2453600, DOI 10.1007/s00222-008-0140-z
- James E. Humphreys, Introduction to Lie algebras and representation theory, Springer-Verlag, New York-Berlin, 1972. Graduate Texts in Mathematics, Vol. 9. MR 0323842
- J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Math., vol. 29, CUP, Cambridge, 1990.
- Alexander S. Kleshchev and Pham Huu Tiep, Representations of the general linear groups which are irreducible over subgroups, Amer. J. Math. 132 (2010), no. 2, 425–473. MR 2654779, DOI 10.1353/ajm.0.0108
- R. Lawther, Jordan block sizes of unipotent elements in exceptional algebraic groups, Comm. Algebra 23 (1995), no. 11, 4125–4156. MR 1351124, DOI 10.1080/00927879508825454
- Martin W. Liebeck and Gary M. Seitz, On the subgroup structure of classical groups, Invent. Math. 134 (1998), no. 2, 427–453. MR 1650328, DOI 10.1007/s002220050270
- 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
- F. Lübeck, Online data for finite groups of Lie type and related algebraic groups, http://www.math.rwth-aachen.de/$\sim$Frank.Luebeck
- 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
- A. A. Premet, Weights of infinitesimally irreducible representations of Chevalley groups over a field of prime characteristic, Mat. Sb. (N.S.) 133(175) (1987), no. 2, 167–183, 271 (Russian); English transl., Math. USSR-Sb. 61 (1988), no. 1, 167–183. MR 905003, DOI 10.1070/SM1988v061n01ABEH003200
- Gary M. Seitz, The maximal subgroups of classical algebraic groups, Mem. Amer. Math. Soc. 67 (1987), no. 365, iv+286. MR 888704, DOI 10.1090/memo/0365
- Stephen D. Smith, Irreducible modules and parabolic subgroups, J. Algebra 75 (1982), no. 1, 286–289. MR 650422, DOI 10.1016/0021-8693(82)90076-X
- Robert Steinberg, Lectures on Chevalley groups, Yale University, New Haven, Conn., 1968. Notes prepared by John Faulkner and Robert Wilson. MR 0466335
- Donna M. Testerman, Irreducible subgroups of exceptional algebraic groups, Mem. Amer. Math. Soc. 75 (1988), no. 390, iv+190. MR 961210, DOI 10.1090/memo/0390