manuscripta mathematica

, Volume 31, Issue 1–3, pp 317–329

Binary polyhedral groups and Euclidean diagrams

  • Dieter Happel
  • Udo Preiser
  • Claus Michael Ringel


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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  1. [1]
    Arnold, V.: The A-D-E-classifications. In: Mathematical developments arising from Hilbert problems (ed. F. E. Browder). Proceedings Symposia Pure Math. vol28, Providence (1976), 46Google Scholar
  2. [2]
    Berman, S; Moody, R.; Wonenburger, M.: Cartan matrices with null roots and finite Cartan matrices. Indiana Math. J.21 (1972), 1091–1099Google Scholar
  3. [3]
    Dornhoff, L.: Group representation theory, vol. A. Marcel Dekker, New York 1971Google Scholar
  4. [4]
    Happel, D.; Preiser, U.; Ringel, C. M.: Vinberg's characterization of Dynkin Diagrams using subadditive functions with application to D Tr—periodic modules. To appear in Proc. Ottawa Conf. Representation Theory of Algebras (1979). Springer Lecture NotesGoogle Scholar
  5. [5]
    Huppert, B.: Endliche Gruppen I. Springer, Berlin 1967Google Scholar
  6. [6]
    Klein, F.: Vorlesungen über das Ikosaeder und die Auflösungen der Gleichungen vom fünften Grade. Leipzig, Teubner, 1884Google Scholar
  7. [7]
    McKay, J.: Affine diagrams and character tables. To appearGoogle Scholar
  8. [8]
    Springer, T. A.: Invariant theory. Springer Lecture Notes in Mathematics.585 (1977)Google Scholar
  9. [9]
    Weber, H.: Lehrbuch der Algebra, Band II (2. Auflage) Braunschweig 1899Google Scholar
  10. [10]
    Vinberg, E. B.: Discrete linear groups generated by reflections. Izv. Akad. Nauk SSSR. Ser. Mat.35(1971), Engl. translation: Math. USSR Izv.5(1971), 1083–1119Google Scholar

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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