McKay’s correspondence and characters of finite subgroups of SU(2)

  • W. Rossmann
Part of the Progress in Mathematics book series (PM, volume 220)


According to MacKay [1980] the irreducible characters of finite sub-groups of SU(2) are in a natural 1-1 correspondence with the extended Coxeter-Dynkin graphs of type ADE. We show that the character values themselves can be given by an uniform formula, as special values of polynomials which arise naturally as numerators of Poincaré series associated to finite subgroups of SU(2) acting on polynomials in two variables. These polynomials have been the subject of a number of investigations, but their interpretation as characters has apparently not been noticed.


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  1. [1]
    N. Bourbaki, Algèbre commutative. Chapitres 1 et 2. Hermann, Paris, 1961.Google Scholar
  2. [2]
    N. Bourbaki, Groupes et algèbres de Lie. Chapitres 4, 5 et 6. Hermann, Paris, 1968.Google Scholar
  3. [3]
    N. Bourbaki, Algèbre commutative. Chapitres 5, 6 et 7. Hermann, Paris, 1985.Google Scholar
  4. [4]
    H.S.M. Coxeter, Regular Complex Polytopes, Second Edition. Cambridge University Press, 1974.Google Scholar
  5. [5]
    G. Gonzales-Sprinberg and J.-L. Verdier, Construction gèomètrique de la correspondence de McKay. Ann. Sci. Ecole Norm. Sup. 16, n°3 (1983), 410–449.Google Scholar
  6. [6]
    E. Hecke, Vorlesungen über die Theorie der Algebraischen Zahlen. Leipzig 1923. Reprinted by Chelsea Publishing Co., New York, 1970.Google Scholar
  7. [7]
    F. Klein, Vorlesungen über das Ikosaeder und die Auflösung der Gleichung vom fünften Grade. Teubner, Leipzig, 1884.Google Scholar
  8. [8]
    H. Knörrer, Group representations and the resolution of rational double points. In: Finite groups - Coming of Age. Proceedings, Montreal 1982 (J. Mckay, ed.). Contemporary Math., v.45, AMS, Providence, 1985, pp. 175–222.Google Scholar
  9. [9]
    B. Kostant, The McKay correspondence, the Coexeter element, and representation theory. In: Élie Cartan et les mathématiques d’aujourd’hui (Lyon, 1984). Astérisque, Hors séries, 1985, pp. 209–255.Google Scholar
  10. [10]
    G. Lusztig, Some examples of square integrable representations of semisimple p-adic groups. Trans. AMS 277 (1983), 153–215.MathSciNetGoogle Scholar
  11. [11]
    G. Lusztig, Subregular nilpotent elements and bases in K-theory. Canad. J. Math. 51 (6) (1999), 1194–1225.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    J. McKay, Graphs, singularities, and finite groups. AMS, Proc. Symp. Pure Math, Vol. 37,1980, pp. 183–186.MathSciNetGoogle Scholar
  13. [13]
    J.P. Serre, Linear Represenations of Finite Groups. Springer Verlag, New York, 1977.CrossRefGoogle Scholar
  14. [14]
    G.C. Shephard and J.A. Todd, Finite unitary reflection groups. Canad. J. Math. 6 (1954), 111–135.MathSciNetGoogle Scholar
  15. [15]
    T.A. Springer, Poincaré series of binary polyhedral groups and McKay’s correspondence. Math. Ann. 278 (1985), 587–598.MathSciNetGoogle Scholar
  16. [16]
    R. Steinberg, Finite subgroups of SU2, affineDynkin diagrams and affine Coxeter elements. Pac. J. Math. 118 (1985), 587–598. Preprint 1982.zbMATHGoogle Scholar
  17. [17]
    H. Weyl, The Classical Groups. Their invariants and Representations, Second Edition. Princeton University Press, Princeton, 1939.Google Scholar
  18. [18]
    O. Zariski and P. Samuel, Commutative Algebra, I. D. Van Nostrand Company, Inc. Princeton, 1958.zbMATHGoogle Scholar

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© Springer Science+Business Media New York 2004

Authors and Affiliations

  • W. Rossmann
    • 1
  1. 1.University of OttawaOttawaCanada

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