Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-745bb68f8f-l4dxg Total loading time: 0 Render date: 2025-01-20T17:22:52.912Z Has data issue: false hasContentIssue false

Irreducible Subgroups of Simple Algebraic Groups – A Survey

Published online by Cambridge University Press:  15 April 2019

By
Timothy C. Burness  and
Donna M. Testerman
Edited by
C. M. Campbell ,
C. W. Parker ,
M. R. Quick ,
E. F. Robertson  and
C. M. Roney-Dougal
C. M. Campbell
Affiliation:
University of St Andrews, Scotland
C. W. Parker
Affiliation:
University of Birmingham
M. R. Quick
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University of St Andrews, Scotland
C. M. Roney-Dougal
Affiliation:
University of St Andrews, Scotland
Get access

Summary

Let G be a simple linear algebraic group over an algebraically closed field K of characteristic p≥ 0, let H be a proper closed subgroup of G and let V be a nontrivial finite dimensional irreducible rational KG-module. We say that (G,H, V) is an irreducible triple if V is irreducible as a KH-module. Determining these triples is a fundamental problem in the representation theory of algebraic groups, which arises naturally in the study of the subgroup structure of classical groups. In the 1980s, Seitz and Testerman extended earlier work of Dynkin on connected subgroups in characteristic zero to all algebraically closed fields. In this article we will survey recent advances towards a classification of irreducible triples for all positive dimensional subgroups of simple algebraic groups.

Type
Chapter
Information
Groups St Andrews 2017 in Birmingham , pp. 230 - 260
Publisher: Cambridge University Press
Print publication year: 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aschbacher, M., On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984), 469–514.CrossRefGoogle Scholar
Borel, A. and Tits, J., Éléments unipotents et sous-groupes paraboliques de groupes réductifs. I, Invent. Math. 12 (1971), 95–104.CrossRefGoogle Scholar
Bourbaki, N., Lie Groups and Lie Algebras (Chapters 4–6), Springer, 2002.CrossRefGoogle Scholar
Burness, T. C., Ghandour, S., Marion, C. and Testerman, D. M., Irreducible almost simple subgroups of classical algebraic groups, Mem. Amer. Math. Soc. 1114, 2015.Google Scholar
Burness, T. C., Ghandour, S. and Testerman, D. M., Irreducible geometric subgroups of classical algebraic groups, Mem. Amer. Math. Soc. 1130, 2016.Google Scholar
Burness, T. C., Marion, C. and Testerman, D. M., On irreducible subgroups of simple algebraic groups, Math. Ann. 367 (2017), 1259–1309.CrossRefGoogle Scholar
Cavallin, M. and Testerman, D. M., A new family of irreducible subgroups of the orthogonal algebraic groups, Trans. Amer. Math. Soc., to appear.Google Scholar
Curtis, C. W., Representations of Lie algebras of classical type with applications to linear groups, J. Math. Mech. 9 (1960), 307–326.Google Scholar
Dynkin, E., Semisimple subalgebras of semisimple Lie algebras, Mat. Sbornik N.S. 30 (72) (1952), 349–462.Google Scholar
Dynkin, E., Maximal subgroups of the classical groups, Amer. Math. Soc. Translations 6 (1957), 245–378.CrossRefGoogle Scholar
Ford, B., Overgroups of irreducible linear groups, I, J. Algebra 181 (1996), 26–69.CrossRefGoogle Scholar
Ford, B., Overgroups of irreducible linear groups, II, Trans. Amer. Math. Soc. 351 (1999), 3869–3913.CrossRefGoogle Scholar
Ford, B. and Kleshchev, A. S., A proof of the Mullineux conjecture, Math. Z. 226 (1997), 267–308.CrossRefGoogle Scholar
Garibaldi, S. and Nakano, D. K., Bilinear and quadratic forms on rational modules of split reductive groups, Canad. J. Math. 68 (2016), 395–421.CrossRefGoogle Scholar
Ghandour, S., Irreducible disconnected subgroups of exceptional algebraic groups, J. Algebra 323 (2010), 2671–2709.CrossRefGoogle Scholar
Hogeweij, G. M. D., Almost-classical Lie algebras, I, Nederl. Akad. Wetensch. Indag. Math. 44 (1982), 441–452.Google Scholar
Jantzen, J. C., Representations of Algebraic Groups, Math. Surveys and Monographs Vol. 107, Amer. Math. Soc., Providence, R.I., 2003.Google Scholar
Liebeck, M. W. and Seitz, G. M., On the subgroup structure of the classical groups, Invent. Math. 134 (1998), 427–453.CrossRefGoogle Scholar
Liebeck, M. W. and Seitz, G. M., The maximal subgroups of positive dimension in exceptional algebraic groups, Mem. Amer. Math. Soc. 802, 2004.Google Scholar
Lübeck, F., Small degree representations of finite Chevalley groups in defining characteristic, LMS J. Comput. Math. 4 (2001), 135–169.CrossRefGoogle Scholar
Malle, G. and Testerman, D., Linear Algebraic Groups and Finite Groups of Lie Type, Cambridge Advanced Studies in Mathematics 133, Cambridge University Press, 2011.CrossRefGoogle Scholar
Premet, A. A. and Suprunenko, I. D., The Weyl modules and the irreducible representations of the symplectic group with the fundamental highest weights, Comm. Algebra 11 (1983), 1309–1342.CrossRefGoogle Scholar
Seitz, G. M., The maximal subgroups of classical algebraic groups, Mem. Amer. Math. Soc. 365, 1987.Google Scholar
Sin, P. and Willems, W., G-invariant quadratic forms, J. Reine Angew. Math. 420 (1991), 45–59.Google Scholar
Smith, S., Irreducible modules and parabolic subgroups, J. Algebra 75 (1982), 286–289.CrossRefGoogle Scholar
Testerman, D. M., Irreducible subgroups of exceptional algebraic groups, Mem. Amer. Math. Soc. 390, 1988.Google Scholar
Testerman, D. M., A construction of certain maximal subgroups of the algebraic groups E6 and F4, J. Algebra 122 (1989), 299–322.CrossRefGoogle Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×