Finite subgroups of \(SU_ 2\), Dynkin diagrams and affine Coxeter elements
type: Item
instance of: scholarly article
title: Finite subgroups of \(SU_ 2\), Dynkin diagrams and affine Coxeter elements
zbMATH Open document ID: 0567.20026
author: Robert Steinberg
published in: Pacific Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: As observed empirically by \textit{J. McKay} [Proc. Symp. Pure Math. 37, 183-186 (1980; Zbl 0451.05026)] the irreducible representations of a finite subgroup of SU(2,\({\mathbb{C}})\) admit a natural parametrization by the nodes of an extended Dynkin diagram of type \(A_ r\), \(D_ r\), or \(E_ r\). In the first part of this article, the author gives a uniform derivation for this result using elementary facts from representation theory and the characterization of root lattices. To see that finite subgroups of SU(2,\({\mathbb{C}})\) and the mentioned Dynkin diagrams correspond bijectively that way, the author invokes a property of Coxeter elements for affine root systems. Some more properties of these elements, among them two normal forms as screw displacements, are derived in the second part of the paper.
Mathematics Subject Classification ID: 20G05
zbMATH DE Number: 3904836
zbMATH Keywords: irreducible representations
zbMATH Keywords: finite subgroup of SU(2,\({\mathbb{C}})\)
zbMATH Keywords: extended Dynkin diagram
zbMATH Keywords: root lattices
zbMATH Keywords: Coxeter elements
zbMATH Keywords: affine root systems
MaRDI profile type: Publication
full work available at URL: pjm.1985.118.587
OpenAlex ID: W2049584888
DOI: 10.2140/PJM.1985.118.587
description: scientific article; zbMATH DE number 3904836