Irreducible representations of simple algebraic groups in which a\n unipotent element is represented by a matrix with single non-trivial Jordan\n block

2016/12/31 by Alexandre Zalesski, Zalesski, Alexandre, Donna Testerman +1
Mathematics · #20G05 #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.1701.00125

openalex publication_date 2016/12/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we prove the following result.\n Let G be a simply connected simple linear algebraic group of exceptional\nLie type over an algebraically closed field F of characteristic p\≥ 0,\nand let u\∈ G be a nonidentity unipotent element. Let \φ be a\nnon-trivial irreducible representation of G.\n Then the Jordan normal form of \φ(u) contains at most one non-trivial\nblock if and only if G is of type G₂, u is a regular unipotent element\nand \dim \φ\≤ 7.\n Note that the irreducible representations of the simple classical algebraic\ngroups in which a non-trivial unipotent element is represented by a matrix\nwhose Jordan form has a single non-trivial block were determined by I.D.\nSuprunenko (Unipotent elements of non-prime order in representations of the\nclassical algebraic groups: two big Jordan blocks, J. Math. Sci. 199(2014), 350\n-- 374.\n

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