×

Jordan block sizes of unipotent elements in exceptional algebraic groups. (English) Zbl 0880.20034

Commun. Algebra 23, No. 11, 4125-4156 (1995); erratum ibid. 26, No. 8, 2709 (1998).
Jordan block sizes of unipotent elements in small representations of the exceptional algebraic groups in positive characteristic \(p\) are determined in the article under review. This is done for two classes of representations: the nontrivial irreducible representations of the minimal dimension and the representations in the Lie algebras of the relevant groups (for the group of type \(E_8\) these classes coincide). The \(p\)th power map on unipotent classes of the exceptional groups is also determined. The main results are represented in 9 tables.
They have been used by M. W. Liebeck, J. Saxl and D. M. Testerman [Proc. Lond. Math. Soc., III. Ser. 72, No. 2, 425-457 (1996; Zbl 0855.20040)] for determining simple subgroups of large rank in simple algebraic groups, by R. Lawther and D. M. Testerman [\(A_1\) subgroups of exceptional algebraic groups (to appear)] for identifying unipotent classes meeting \(A_1\) subgroups of the exceptional algebraic groups and by the reviewer [in Proc. Lond. Math. Soc., III. Ser. 71, No. 2, 281-332 (1995; Zbl 0835.20057)] for classifying irreducible representations of simple algebraic groups which contain matrices with large Jordan blocks.

MSC:

20G05 Representation theory for linear algebraic groups
20G40 Linear algebraic groups over finite fields
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1112/plms/s3-46.1.38 · Zbl 0503.20013 · doi:10.1112/plms/s3-46.1.38
[2] DOI: 10.1017/S0305004100052403 · Zbl 0364.22006 · doi:10.1017/S0305004100052403
[3] DOI: 10.1017/S0305004100052610 · Zbl 0364.22007 · doi:10.1017/S0305004100052610
[4] Carter R.W., Simple Groups of Lie Type (1972) · Zbl 0248.20015
[5] Carter R.W., Finite Groups of Lie Type:Conjugacy Classes and Complex Characters (1985)
[6] Chang B., The conjugate classes of Chevalley groups of type (G2)”, J. Algebra 9 pp 190– (1968)
[7] Dynkin E.B., Semisimple subalgebras of semisimple Lie algebras”, Amer. Math. Soc. Trans 6 pp 111– (1957) · Zbl 0077.03404
[8] Enomoto H., The conjugacy classes of Chevalley groups of type (G2) over finite fields of characteristic 2 or 3”, J. Fac. Sci. Univ. Tokyo 16 pp 497– (1970) · Zbl 0242.20049
[9] Lawther R., Ai subgroups of exceptional algebraic groups · Zbl 0936.20039
[10] Liebeck M. W., Simple subgroups of large rank in groups of Lie type · Zbl 0855.20040
[11] Mizuno K., The conjugate classes of Chevalley groups of type EG” , J. Fac. Sci. Univ. Tokyo 24 pp 525– (1977) · Zbl 0399.20044
[12] Mizuno K., The conjugate classes of the Chevalley groups of type E7 and E&”, Tokyo J. Math 3 pp 391– (1980)
[13] Pommerening K., über die unipotenten Klassen reduktiver Gruppen J. Algebra 49 pp 525– (1980) · Zbl 0437.20034
[14] Pommerening K., Uber die unipotenten Klassen reduktiver Gruppen II”, J. Algebra 65 pp 373– (1980) · Zbl 0437.20034
[15] DOI: 10.1112/blms/6.1.21 · Zbl 0287.20036 · doi:10.1112/blms/6.1.21
[16] Shinoda K., The conjugacy classes of Chevalley groups of type (F4) over finite fields of characteristic 2”, J. Fac. Sci. Univ. Tokyo 21 pp 133– (1974) · Zbl 0306.20013
[17] Shoji T., The conjugacy classes of Chevalley groups of type (F4) over finite fields of characteristic p ^ 2”, J. Fac. Sci. Univ. Tokyo 21 pp 1– (1974) · Zbl 0279.20038
[18] Slodowy P., Simple singularities and simple algebraic groups 815 (1980) · Zbl 0441.14002 · doi:10.1007/BFb0090294
[19] Stuhler U., Unipotente und nilpotente Klassen in einfachen Gruppen und Lie Algebren von Typ G2”, Indag. Math 33 pp 365– (1971) · Zbl 0232.20085
[20] Suprunenko I., Irreducible representations of simple algebraic groups containing matrices with big Jordan blocks · Zbl 0835.20057 · doi:10.1112/plms/s3-71.2.281
[21] Testerman D.M., Irreducible subgroups of exceptional algebraic groups, Memoirs Amer. Math. Soc 390 (1988) · Zbl 0662.20037
[22] Testerman D.M., A construction of certain maximal subgroups of the algebraic groups E6 and F4’J. Algebra 122 pp 299– (1989)
[23] Testerman D. M., A1-type overgroups of elements of order p in semisimple algebraic groups and the associated finite groups · Zbl 0857.20025 · doi:10.1006/jabr.1995.1285
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.