The centralizers of semisimple elements of the Chevalley groups \(E_ 7\) and \(E_ 8\)
type: Item
instance of: scholarly article
title: The centralizers of semisimple elements of the Chevalley groups \(E_ 7\) and \(E_ 8\)
zbMATH Open document ID: 0534.20031
author: D. I. Deriziotis
published in: Tokyo Journal of Mathematics
publication date: 1983-01-01T00:00:00Z
review text: The purpose of this paper is to give in detailed tables all the centralizers and their orders of semisimple elements of the finite Chevalley groups \(E_ 7\) and \(E_ 8\). (Two tables on 21 pages.) Table 1: The structure and the orders of connected centralizers of semisimple elements in \(E_ 7\) (7 pp.). Table 2: The structure and the orders of the centralizers of semisimple elements in \(E_ 8\) (14 pp.). These tables are very useful since they give the character degrees of the semisimple irreducible complex representations constructed by Deligne and Lusztig for the finite Chevalley groups of adjoint type. These degrees can be obtained if we know what subgroups of the finite Chevalley groups of universal type are centralizers of semisimple elements [cf. \textit{P. Deligne} and \textit{G. Lusztig}, Ann. Math., II. Ser. 103, 103-161 (1976; Zbl 0336.20029)]. Similar tables giving these centralizers for the classical groups and for the groups \(G_ 2\), \(F_ 4\) and \(E_ 6\) have been obtained respectively by [\textit{R. W. Carter}, Proc. Lond. Math. Soc., III. Ser. 42, 1-41 (1981; Zbl 0455.20035)], \textit{B. Chang} and \textit{R. Ree}, Symp. Math. 13, 395-413 (1974; Zbl 0314.20034), \textit{T. Shoji}, J. Fac. Sci., Univ. Tokyo, Sect. IA 21, 1-17 (1974; Zbl 0279.20038), \textit{K. Mizuno}, ibid. 24, 525-563 (1977; Zbl 0399.20044). Cf. also \textit{K. Mizuno}, Tokyo J. Math. 3, 391-459 (1980; Zbl 0454.20046)].
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20G05
zbMATH DE Number: 3847669
zbMATH Keywords: finite Chevalley groups
zbMATH Keywords: connected centralizers of semisimple elements
zbMATH Keywords: character degrees
zbMATH Keywords: semisimple irreducible complex representations
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270214335
OpenAlex ID: W2041443211
DOI: 10.3836/TJM/1270214335
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description: scientific article; zbMATH DE number 3847669