Finite subgroups of \(SU_ 2\), Dynkin diagrams and affine Coxeter elements
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.
finite subgroup of SU(2,\({\mathbb{C}})\)
scientific article; zbMATH DE number 3904836