On conjugacy classes of elements of finite order in complex semisimple Lie groups
type: Item
instance of: scholarly article
title: On conjugacy classes of elements of finite order in complex semisimple Lie groups
zbMATH Open document ID: 0564.22011
DOI: 10.1016/0022-4049(85)90026-X
published in: Journal of Pure and Applied Algebra
publication date: 1985-01-01T00:00:00Z
review text: Let G be a connected semi-simple complex Lie group, and k be an integer. The main theme of this paper is the study of the number \(\nu\) (G,k) of conjugacy classes of G consisting of elements whose orders divide k. The author first shows that \(\nu (G^{\vee},k)=\nu (G,k)\) for the dual group \(G^{\vee}\) of G. Thus the case of adjoint groups reduces to the case of simply connected groups. Then he obtains a simple formula for the generating function \(F(G,t)=\sum_{k\geq 0}\nu (G,k)t^ k\) in the case when G is simply connected. In a previous paper, the author shows that \(\nu (G,k)=\prod^{\ell}_{i=1}(m_ i+k)/(m_ i+1),\) where \(m_ i\) (1\(\leq i\leq \ell)\) are exponents of the Weyl group W of G, under the condition that k and the order of W are relatively prime [Proc. Am. Math. Soc. 80, 181-184 (1980; Zbl 0436.22008)]. He shows that for simply connected almost simple G, this formula is also valid under some weaker conditions. Further, when G is a group of type \(A_{\ell}\), an explicit formula for \(\nu\) (G,k) and also some new partition identities are given.
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 11P81
zbMATH DE Number: 3899198
zbMATH Keywords: semi-simple complex Lie group
zbMATH Keywords: conjugacy classes
zbMATH Keywords: dual group
zbMATH Keywords: Weyl group
zbMATH Keywords: partition identities
author: Dragomir Ž. Đoković
reviewed by: Eiichi Abe
MaRDI profile type: Publication
Wikidata QID: Q115364455
cites work: On the Geometry of Numbers in Elementary Number Theory
cites work: On Conjugacy Classes of Elements of Finite Order in Compact or Complex Semisimple Lie Groups
cites work: Q5734920
cites work: Endliche Gruppen I
cites work: Some properties of the cyclotomic polynomial
cites work: Characters of Elements of Finite Order in Lie Groups
cites work: Q5519950
full work available at URL: 0022-4049(85)90026-x
OpenAlex ID: W2044835537
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description: scientific article; zbMATH DE number 3899198