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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
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