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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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