manuscripta mathematica

, Volume 31, Issue 1–3, pp 317–329

Binary polyhedral groups and Euclidean diagrams

  • Dieter Happel
  • Udo Preiser
  • Claus Michael Ringel
Article

Abstract

Recently, J. McKay [7] has observed that the irreducible complex representations of the binary polyhedral groups can be arranged in order to form the vertices of a Euclidean diagram in such a way that the tensor product of any irreducible representation M with the standard two-dimensional representation is the direct sum of the irreducible representations which are the neighbors of M in the diagram, and he asked for an explanation. In this note, we will show that any self-dual two-dimensional representation gives rise to a generalized Euclidean diagram, and that this in fact can be used to give a proof of the classification theorem of the binary polyhedral groups which at the same time furnishes a list of the irreducible representations and also gives the minimal splitting field.

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Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • Dieter Happel
    • 1
  • Udo Preiser
    • 1
  • Claus Michael Ringel
    • 1
  1. 1.Fakultät für MathematikUniversität BielefeldBielefeld 1

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