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5 - Complete reducibility and subgroups of exceptional algebraic groups

Published online by Cambridge University Press:  21 November 2024

Edited by
C. M. Campbell ,
M. R. Quick ,
E. F. Robertson ,
C. M. Roney-Dougal  and
D. I. Stewart
C. M. Campbell
Affiliation:
University of St Andrews, Scotland
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
D. I. Stewart
Affiliation:
University of Manchester
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Summary

This survey article has two components. The first part gives a gentle introduction to Serres notion of $G$-complete reducibility, where $G$ is a connected reductive algebraic group defined over an algebraically closed field. The second part concerns consequences of this theory when $G$ is simple of exceptional type, specifically its role in elucidating the subgroup structure of $G$. The latter subject has a history going back about sixty years. We give an overview of what is known, up to the present day. We also take the opportunity to offer several corrections to the literature.

Keywords

algebraic groupexceptional groupcomplete reducibility
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Groups St Andrews 2022 in Newcastle , pp. 191 - 245
Publisher: Cambridge University Press
Print publication year: 2024

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References

[1] Amende, Bonnie. 2005. G-irreducible subgroups of type A1. ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)–University of Oregon.Google Scholar
[2] Aschbacher, M., and Scott, L. 1985. Maximal subgroups of finite groups. J. Algebra, 92(1), 44–80.CrossRefGoogle Scholar
[3] Aschbacher, Michael. 1987. The 27-dimensional module for E6. I. Invent. Math., 89(1), 159–195.CrossRefGoogle Scholar
[4] Aschbacher, Michael. 1988. The 27-dimensional module for E6. II. J. London Math. Soc. (2), 37(2), 275–293.CrossRefGoogle Scholar
[5] Aschbacher, Michael. 1990a. The 27-dimensional module for E6. III. Trans. Amer. Math. Soc., 321(1), 45–84.Google Scholar
[6] Aschbacher, Michael. 1990b. The 27-dimensional module for E6. IV. J. Algebra, 131(1), 23–39.CrossRefGoogle Scholar
[7] Attenborough, Christopher, Bate, Michael, Gruchot, Maike, Litterick, Alastair, and Röhrle, Gerhard. 2020. On relative complete reducibility. Q. J. Math., 71(1), 321–334.CrossRefGoogle Scholar
[8] Azad, H., Barry, M., and Seitz, G. 1990. On the structure of parabolic subgroups. Comm. Algebra, 18(2), 551–562.CrossRefGoogle Scholar
[9] Ballantyne, John, Bates, Chris, and Rowley, Peter. 2015. The maximal subgroups of E7(2). LMS J. Comput. Math., 18(1), 323–371.CrossRefGoogle Scholar
[10] Bate, M., Martin, B., Röhrle, G., and Tange, R. 2011. Complete reducibility and conjugacy classes of tuples in algebraic groups and Lie algebras. Math. Z., 269(3-4), 809–832.CrossRefGoogle Scholar
[11] Bate, Michael, Martin, Benjamin, and Röhrle, Gerhard. 2005. A geometric approach to complete reducibility. Invent. Math., 161(1), 177–218.CrossRefGoogle Scholar
[12] Bate, Michael, Martin, Benjamin, and Röhrle, Gerhard. 2008. Complete reducibility and commuting subgroups. J. Reine Angew. Math., 621, 213–235.Google Scholar
[13] Bate, Michael, Martin, Benjamin, Röhrle, Gerhard, and Tange, Rudolf. 2013. Closed orbits and uniform S-instability in geometric invariant theory. Trans. Amer. Math. Soc., 365(7), 3643–3673.CrossRefGoogle Scholar
[14] Bate, Michael, Martin, Benjamin, and Röhrle, Gerhard. 2022. Overgroups of regular unipotent elements in reductive groups. Forum Math. Sigma, 10, Paper No. e13, 13.Google Scholar
[15] Bendel, C. P., Nakano, D. K., and Pillen, C. 2004. Extensions for Frobenius kernels. J. Algebra, 272(2), 476–511.CrossRefGoogle Scholar
[16] Bendel, Christopher P., Nakano, Daniel K., Parshall, Brian J., Pillen, Cornelius, Scott, Leonard L., and Stewart, David. 2015. Bounding cohomology for finite groups and Frobenius kernels. Algebr. Represent. Theory, 18(3), 739–760.CrossRefGoogle Scholar
[17] Borel, A. 1991. Linear algebraic groups. Second edn. Graduate Texts in Mathematics, vol. 126. New York: Springer-Verlag.CrossRefGoogle Scholar
[18] Borel, A., and Tits, J. 1971. Éléments unipotents et sous-groupes paraboliques de groupes réductifs. I. Invent. Math., 12, 95–104.CrossRefGoogle Scholar
[19] Borel, A., and Tits, J. 1973. Homomorphismes “abstraits” de groupes algébriques simples. Ann. of Math. (2), 97, 499–571.CrossRefGoogle Scholar
[20] Borovik, A. V. 1989a. Jordan subgroups of simple algebraic groups. Algebra i Logika, 28(2), 144–159, 244.Google Scholar
[21] Borovik, A. V. 1989b. The structure of finite subgroups of simple algebraic groups. Algebra i Logika, 28(3), 249–279, 366.Google Scholar
[22] Bourbaki, N. 2005. Lie groups and Lie algebras. Chapters 7–9. Elements of Mathematics (Berlin). Berlin: Springer-Verlag.Google Scholar
[23] Burness, Timothy C., and Testerman, Donna M. 2019. Irreducible subgroups of simple algebraic groups – a survey. Pages 230–260 of: Groups St Andrews 2017 in Birmingham. London Math. Soc. Lecture Note Ser., vol. 455. Cambridge Univ. Press, Cambridge.CrossRefGoogle Scholar
[24] Caprace, Pierre-Emmanuel. 2009. “Abstract” homomorphisms of split Kac-Moody groups. Mem. Amer. Math. Soc., 198(924), xvi+84.Google Scholar
[25] Carter, Roger W. 1993. Finite groups of Lie type. Wiley Classics Library. Chichester: John Wiley & Sons Ltd. Conjugacy classes and complex characters, Reprint of the 1985 original, A Wiley-Interscience Publication.Google Scholar
[26] Chevalley, Claude. 1951. Théorie des groupes de Lie. Tome II. Groupes algébriques. Actualités Sci. Ind. no. 1152. Hermann & Cie., Paris.Google Scholar
[27] Cline, E., Parshall, B., Scott, L., and van der Kallen, W. 1977. Rational and generic cohomology. Invent. Math., 39(2), 143–163.CrossRefGoogle Scholar
[28] Cohen, Arjeh M., and Wales, David B. 1995. Finite simple subgroups of semisimple complex Lie groups – a survey. Pages 77–96 of: Groups of Lie type and their geometries (Como, 1993). London Math. Soc. Lecture Note Ser., vol. 207. Cambridge: Cambridge Univ. Press.CrossRefGoogle Scholar
[29] Cooperstein, Bruce N. 1981. Maximal subgroups of G2(2n). J. Algebra, 70(1), 23–36.CrossRefGoogle Scholar
[30] Craven, David A. 2017. Alternating subgroups of exceptional groups of Lie type. Proc. Lond. Math. Soc. (3), 115(3), 449–501.CrossRefGoogle Scholar
[31] Craven, David A. 2021. The maximal subgroups of the exceptional groups F4(q), E6(q) and 2E6(q) and related almost simple groups. arXiv:2103.04869.Google Scholar
[32] Craven, David A. 2022a. Maximal PSL2 subgroups of exceptional groups of Lie type. Mem. Amer. Math. Soc., 276(1355), v+155.Google Scholar
[33] Craven, David A. 2022b. On the maximal subgroups of E7(q) and related almost simple groups. arXiv:2201.07081.Google Scholar
[34] Craven, David A. to appear. On medium-rank Lie primitive and maximal subgroups of exceptional groups of Lie type. Mem. Amer. Math. Soc.Google Scholar
[35] Craven, David A., Stewart, David I., and Thomas, Adam R. 2022. A new maximal subgroup of E8 in characteristic 3. Proc. Amer. Math. Soc., 150(4), 1435–1448.CrossRefGoogle Scholar
[36] Dawson, Denise Karin. 2011. Complete reducibility in Euclidean twin buildings. ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)–Cornell University.Google Scholar
[37] Dowd, Michael F., and Sin, Peter. 1996. On representations of algebraic groups in characteristic two. Comm. Algebra, 24(8), 2597–2686.CrossRefGoogle Scholar
[38] Dynkin, E. B. 2000. Selected papers of E. B. Dynkin with commentary. American Mathematical Society, Providence, RI; International Press, Cambridge, MA. Edited by Yushkevich, A. A., Seitz, G. M. and Onishchik, A. L..Google Scholar
[39] Fulton, William, and Harris, Joe. 1991. Representation theory, A first course. Graduate Texts in Mathematics, vol. 129. New York: Springer-Verlag.Google Scholar
[40] Ganeshalingam, Vanthana, and Thomas, Adam R. On the noncompletely reducible subgroups of F4. In preparation.Google Scholar
[41] Goodwin, Simon M., and Pengelly, Rachel. On sl2-triples for classical algebraic groups in positive characteristic. Transform. Groups, to appear.Google Scholar
[42] Gorenstein, Daniel, Lyons, Richard, and Solomon, Ronald. 1998. The classification of the finite simple groups. Number 3. Part I. Chapter A. Mathematical Surveys and Monographs, vol. 40. American Mathematical Society, Providence, RI.Google Scholar
[43] Gruchot, Maike, Litterick, Alastair, and Röhrle, Gerhard. 2020. Relative complete reducibility and normalized subgroups. Forum Math. Sigma, 8, Paper No. e30, 32.Google Scholar
[44] Gruchot, Maike, Litterick, Alastair, and Röhrle, Gerhard. 2022. Complete reducibility: variations on a theme of Serre. Manuscripta Math., 168(3-4), 439–451.CrossRefGoogle Scholar
[45] Herpel, Sebastian. 2013. On the smoothness of centralizers in reductive groups. Trans. Amer. Math. Soc., 365(7), 3753–3774.CrossRefGoogle Scholar
[46] Herpel, Sebastian, and Stewart, David I. 2016a. Maximal subalgebras of Cartan type in the exceptional Lie algebras. Selecta Math., 22(2), 765–799.CrossRefGoogle Scholar
[47] Herpel, Sebastian, and Stewart, David I. 2016b. On the smoothness of normalisers, the subalgebra structure of modular Lie algebras, and the cohomology of small representations. Doc. Math., 21, 1–37.CrossRefGoogle Scholar
[48] Herpel, Sebastian, Röhrle, Gerhard, and Gold, Daniel. 2011. Complete reducibility and Steinberg endomorphisms. C. R. Math. Acad. Sci. Paris, 349(5-6), 243–246.CrossRefGoogle Scholar
[49] Humphreys, James E. 1975. Linear algebraic groups. New York: Springer-Verlag. Graduate Texts in Mathematics, No. 21.CrossRefGoogle Scholar
[50] Humphreys, James E. 1978. Introduction to Lie algebras and representation theory. Graduate Texts in Mathematics, vol. 9. Springer-Verlag, New York-Berlin. Second printing, revised.Google Scholar
[51] Humphreys, James E. 1990. Reflection groups and Coxeter groups. Cambridge Studies in Advanced Mathematics, vol. 29. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
[52] Jacobson, Nathan. 1951. Completely reducible Lie algebras of linear transformations. Proc. Amer. Math. Soc., 2, 105–113.CrossRefGoogle Scholar
[53] Jantzen, J. C. 1991. First cohomology groups for classical Lie algebras. Pages 289–315 of: Representation theory of finite groups and finitedimensional algebras (Bielefeld, 1991). Progr. Math., vol. 95. Basel: Birkhäuser.CrossRefGoogle Scholar
[54] Jantzen, J. C. 1997. Low-dimensional representations of reductive groups are semisimple. Pages 255–266 of: Algebraic groups and Lie groups. Austral. Math. Soc. Lect. Ser., vol. 9. Cambridge Univ. Press, Cambridge.Google Scholar
[55] Jantzen, J. C. 2003. Representations of algebraic groups. Second edn. Mathematical Surveys and Monographs, vol. 107. Providence, RI: American Mathematical Society.Google Scholar
[56] Kleidman, Peter. 1988. The maximal subgroups of the Steinberg triality groups 3D4(q) and of their automorphism groups. J. Algebra, 115(1), 182–199.CrossRefGoogle Scholar
[57] Kleidman, Peter, and Liebeck, Martin. 1990. The subgroup structure of the finite classical groups. London Mathematical Society Lecture Note Series, vol. 129. Cambridge: Cambridge University Press.Google Scholar
[58] Kostant, Bertram. 1959. The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group. Amer. J. Math., 81, 973–1032.CrossRefGoogle Scholar
[59] Lawther, R. 2009. Unipotent classes in maximal subgroups of exceptional algebraic groups. J. Algebra, 322(1), 270–293.CrossRefGoogle Scholar
[60] Lawther, R., and Testerman, D. M. 1999. A1 Subgroups of Exceptional Algebraic Groups. Mem. Amer. Math. Soc., 141(674).Google Scholar
[61] Lie, Sophus. 1880. Theorie der Transformationsgruppen I. Math. Ann., 16(4), 441–528.CrossRefGoogle Scholar
[62] Liebeck, Martin W., and Seitz, Gary M. 1994. Subgroups generated by root elements in groups of Lie type. Ann. of Math. (2), 139(2), 293–361.CrossRefGoogle Scholar
[63] Liebeck, Martin W., and Seitz, Gary M. 1996. Reductive subgroups of exceptional algebraic groups. Mem. Amer. Math. Soc., 121(580), vi+111.Google Scholar
[64] Liebeck, Martin W., and Seitz, Gary M. 1998. On the subgroup structure of exceptional groups of Lie type. Trans. Amer. Math. Soc., 350(9), 3409–3482.CrossRefGoogle Scholar
[65] Liebeck, Martin W., and Seitz, Gary M. 1999. On finite subgroups of exceptional algebraic groups. J. Reine Angew. Math., 515, 25–72.CrossRefGoogle Scholar
[66] Liebeck, Martin W., and Seitz, Gary M. 2003. A survey of maximal subgroups of exceptional groups of Lie type. Pages 139–146 of: Groups, combinatorics & geometry (Durham, 2001). World Sci. Publ., River Edge, NJ.CrossRefGoogle Scholar
[67] Liebeck, Martin W., and Seitz, Gary M. 2004. The maximal subgroups of positive dimension in exceptional algebraic groups. Mem. Amer. Math. Soc., 169(802), vi+227.Google Scholar
[68] Liebeck, Martin W., and Seitz, Gary M. 2012. Unipotent and nilpotent classes in simple algebraic groups and Lie algebras. Mathematical Surveys and Monographs, vol. 180. American Mathematical Society, Providence, RI.Google Scholar
[69] Liebeck, Martin W., and Testerman, Donna M. 2004. Irreducible subgroups of algebraic groups. Q. J. Math., 55(1), 47–55.CrossRefGoogle Scholar
[70] Liebeck, Martin W., and Thomas, Adam R. 2017. Finite subgroups of simple algebraic groups with irreducible centralizers. J. Group Theory, 20(5), 841–870.CrossRefGoogle Scholar
[71] Liebeck, Martin W., Saxl, Jan, and Seitz, Gary M. 1996. Factorizations of simple algebraic groups. Trans. Amer. Math. Soc., 348(2), 799–822.CrossRefGoogle Scholar
[72] Liebeck, Martin W., Martin, Benjamin M. S., and Shalev, Aner. 2005. On conjugacy classes of maximal subgroups of finite simple groups, and a related zeta function. Duke Math. J., 128(3), 541–557.CrossRefGoogle Scholar
[73] Litterick, Alastair J. 2018. On non-generic finite subgroups of exceptional algebraic groups. Mem. Amer. Math. Soc., 253(1207), v+156.Google Scholar
[74] Litterick, Alastair J., and Thomas, Adam R. 2018a. Complete reducibility in good characteristic. Trans. Amer. Math. Soc., 370(8), 5279–5340.CrossRefGoogle Scholar
[75] Litterick, Alastair J., and Thomas, Adam R. 2018b. Reducible subgroups of exceptional algebraic groups. Journal of Pure and Applied Algebra, 2489–2529.Google Scholar
[76] Magaard, Kay. 1990. The maximal subgroups of the Chevalley groups F(,4)(F) where F is a finite or algebraically closed field of characteristic not equal to 2,3. ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)– California Institute of Technology.Google Scholar
[77] Malle, Gunter. 1991. The maximal subgroups of 2F4(q2). J. Algebra, 139(1), 52–69.CrossRefGoogle Scholar
[78] Malle, Gunter, and Testerman, Donna. 2011. Linear algebraic groups and finite groups of Lie type. Cambridge Studies in Advanced Mathematics, vol. 133. Cambridge: Cambridge University Press.Google Scholar
[79] Malle, Gunter, and Testerman, Donna. 2021. Overgroups of regular unipotent elements in simple algebraic groups. Trans. Amer. Math. Soc. Ser. B, 8, 788–822.CrossRefGoogle Scholar
[80] McNinch, George J. 1998. Dimensional criteria for semisimplicity of representations. Proc. London Math. Soc. (3), 76(1), 95–149.CrossRefGoogle Scholar
[81] McNinch, George J. 2007. Completely reducible Lie subalgebras. Transformation Groups, 12(1), 127–135.CrossRefGoogle Scholar
[82] McNinch, George J. 2014. Linearity for actions on vector groups. J. Algebra, 397, 666–688.CrossRefGoogle Scholar
[83] McNinch, George J., and Testerman, Donna M. 2007. Completely reducible SL(2)-homomorphisms. Trans. Amer. Math. Soc., 359(9), 4489–4510.CrossRefGoogle Scholar
[84] Milne, J. S. 2017. Algebraic groups. The theory of group schemes of finite type over a field. Vol. 170. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
[85] Morozov, V. V. 1942. On a nilpotent element in a semi-simple Lie algebra. C. R. (Doklady) Acad. Sci. URSS (N.S.), 36, 83–86.Google Scholar
[86] Parker, Alison E. 2007. Higher extensions between modules for SL2. Adv. Math., 209(1), 381–405.CrossRefGoogle Scholar
[87] Pommerening, Klaus. 1980. Über die unipotenten Klassen reduktiver Gruppen. II. J. Algebra, 65(2), 373–398.CrossRefGoogle Scholar
[88] Prasad, Gopal, and Yu, Jiu-Kang. 2006. On quasi-reductive group schemes. J. Algebraic Geom., 15(3), 507–549. With an appendix by Brian Conrad.CrossRefGoogle Scholar
[89] Premet, Alexander. 2017. A modular analogue of Morozov’s theorem on maximal subalgebras of simple Lie algebras. Adv. Math., 311, 833–884.CrossRefGoogle Scholar
[90] Premet, Alexander, and Stewart, David I. 2019. Classification of the maximal subalgebras of exceptional Lie algebras over fields of good characteristic. J. Amer. Math. Soc., 32(4), 965–1008.CrossRefGoogle Scholar
[91] Premet, Alexander, and Strade, Helmut. 2006. Classification of finite dimensional simple Lie algebras in prime characteristics. Pages 185–214 of: Representations of algebraic groups, quantum groups, and Lie algebras. Contemp. Math., vol. 413. Amer. Math. Soc., Providence, RI.CrossRefGoogle Scholar
[92] Richardson, R. W. 1967. Conjugacy classes in Lie algebras and algebraic groups. Ann. of Math. (2), 86, 1–15.CrossRefGoogle Scholar
[93] Richardson, R. W. 1977. Affine coset spaces of reductive algebraic groups. Bull. London Math. Soc., 9(1), 38–41.CrossRefGoogle Scholar
[94] Richardson, R. W. 1988. Conjugacy classes of n-tuples in Lie algebras and algebraic groups. Duke Math. J., 57(1), 1–35.CrossRefGoogle Scholar
[95] Riche, Simon, and Williamson, Geordie. 2021. A simple character formula. Ann. H. Lebesgue, 4, 503–535.CrossRefGoogle Scholar
[96] Saxl, Jan, and Seitz, Gary M. 1997. Subgroups of algebraic groups containing regular unipotent elements. J. London Math. Soc. (2), 55(2), 370–386.CrossRefGoogle Scholar
[97] Seitz, Gary M. 1987. The maximal subgroups of classical algebraic groups. Mem. Amer. Math. Soc., 67(365), iv+286.Google Scholar
[98] Seitz, Gary M. 1991. Maximal subgroups of exceptional algebraic groups. Mem. Amer. Math. Soc., 90(441), iv+197.Google Scholar
[99] Serre, Jean-Pierre. 1994. Sur la semi-simplicité des produits tensoriels de représentations de groupes. Invent. Math., 116(1-3), 513–530.CrossRefGoogle Scholar
[100] Serre, Jean-Pierre. 1997. Semisimplicity and tensor products of group representations: converse theorems. J. Algebra, 194(2), 496–520. With an appendix by Walter Feit.CrossRefGoogle Scholar
[101] Serre, Jean-Pierre. 1998. Morsund lectures, University of Oregon.Google Scholar
[102] Serre, Jean-Pierre. 2005. Complète réductibilité. Astérisque, Exp. No. 932, viii, 195–217. Séminaire Bourbaki. Vol. 2003/2004.Google Scholar
[103] Springer, T. A. 1998. Linear algebraic groups. Second edn. Progress in Mathematics, vol. 9. Boston, MA: Birkhäuser Boston Inc.CrossRefGoogle Scholar
[104] Springer, T. A., and Steinberg, R. 1970. Conjugacy classes. Pages 167–266 of: Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69). Lecture Notes in Mathematics, Vol. 131. Springer, Berlin.CrossRefGoogle Scholar
[105] Stewart, David I. 2010. The reductive subgroups of G2. J. Group Theory, 13(1), 117–130.Google Scholar
[106] Stewart, David I. 2013a. On unipotent algebraic G-groups and 1-cohomology. Trans. Amer. Math. Soc., 365(12), 6343–6365.CrossRefGoogle Scholar
[107] Stewart, David I. 2013b. The reductive subgroups of F4. Mem. Amer. Math. Soc., 223(1049), vi+88.Google Scholar
[108] Stewart, David I. 2014. Non-G-completely reducible subgroups of the exceptional algebraic groups. Int. Math. Res. Not. IMRN, 6053–6078.CrossRefGoogle Scholar
[109] Stewart, David I., and Thomas, Adam R. 2018. The Jacobson-Morozov theorem and complete reducibility of Lie subalgebras. Proc. Lond. Math. Soc. (3), 116(1), 68–100.CrossRefGoogle Scholar
[110] Strade, H. 2004. Simple Lie algebras over fields of positive characteristic. I. de Gruyter Expositions in Mathematics, vol. 38. Berlin: Walter de Gruyter & Co. Structure theory.CrossRefGoogle Scholar
[111] Sury, B. 2016. Hermann Weyl and representation theory. Resonance, 21, 1073–1091.CrossRefGoogle Scholar
[112] Testerman, Donna, and Zalesski, Alexandre. 2013. Irreducibility in algebraic groups and regular unipotent elements. Proc. Amer. Math. Soc., 141(1), 13–28.CrossRefGoogle Scholar
[113] Testerman, Donna M. 1988. Irreducible subgroups of exceptional algebraic groups. Mem. Amer. Math. Soc., 75(390), iv+190.Google Scholar
[114] Testerman, Donna M. 1992. The construction of the maximal A1’s in the exceptional algebraic groups. Proc. Amer. Math. Soc., 116(3), 635–644.Google Scholar
[115] Thomas, Adam R. 2015. Simple irreducible subgroups of exceptional algebraic groups. J. Algebra, 423, 190–238.CrossRefGoogle Scholar
[116] Thomas, Adam R. 2016. Irreducible A1 subgroups of exceptional algebraic groups. J. Algebra, 447, 240–296.CrossRefGoogle Scholar
[117] Thomas, Adam R. 2020. The irreducible subgroups of exceptional algebraic groups. Mem. Amer. Math. Soc., 268(1307), v+191.Google Scholar
[118] Vasiu, A. 2005. Normal, unipotent subgroup schemes of reductive groups. C. R. Math. Acad. Sci. Paris, 341(2), 79–84.CrossRefGoogle Scholar
[119] Weyl, H. 1925. Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen. I. Math. Z.,CrossRefGoogle Scholar

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