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A simplification of the computation of the natural representation of the symmetric group $ S\sb{n}$

Author: Joseph M. Clifton
Journal: Proc. Amer. Math. Soc. 83 (1981), 248-250
MSC: Primary 20C30
MathSciNet review: 624907
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Abstract: Recent use of the symmetric group $ {S_n}$ in processing identities in nonassociative algebras has brought a renewed interest in the natural (integral) irreducible representation of $ {S_n}$ [3]. Using a construction due to A. Young, H. Boerner gives a prescription for writing down the natural (integral) irreducible representation of the symmetric group $ {S_n}$ over an algebraically closed field of characteristic zero [1, p. 119]. Writing down the matrices using this prescription is rather tedious, and becomes computationally impossible for large $ n(n > 9)$ because of the need to calculate chains. In this paper, we simplify the computation of these matrices by eliminating the need to calculate chains. The calculation of the chains is replaced by the simple act of setting up and inverting an upper triangular matrix $ {A_I}$.

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  • [1] Hermann Boerner, Representation of groups with special consideration for the needs of modern physics, Translated from the German by P. G. Murphy in cooperation with J. Mayer-Kalkschmidt and P. Carr, North-Holland Publishing Co., Amsterdam; Interscience Publishers, a division of John Wiley & Sons, Inc., New York; 1963, 1963. MR 0148766
  • [2] J. Clifton, Complete sets of orthogonal tableaux, Ph. D. Dissertation, Iowa State University, Ames, Iowa, 1980.
  • [3] Irvin Roy Hentzel, Processing identities by group representation, Computers in nonassociative rings and algebras (Special session, 82nd Annual Meeting Amer. Math. Soc., San Antonio, Tex., 1976) Academic Press, New York, 1977, pp. 13–40. MR 0463251
  • [4] Daniel Edwin Rutherford, Substitutional Analysis, Edinburgh, at the University Press, 1948. MR 0027272
  • [5] Alfred Young, The collected papers of Alfred Young (1873–1940), University of Toronto Press, Toronto, Ont., Buffalo, N. Y., 1977. With a foreword by G. de B. Robinson and a biography by H. W. Turnbull; Mathematical Expositions, No. 21. MR 0439548

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