A simplification of the computation of the natural representation of the symmetric group 𝑆_{𝑛}

Clifton, Joseph M.

doi:10.1090/S0002-9939-1981-0624907-3

A simplification of the computation of the natural representation of the symmetric group $S_{n}$
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by Joseph M. Clifton PDF

Proc. Amer. Math. Soc. 83 (1981), 248-250 Request permission

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 ${A_I}$.

References

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Complete sets of orthogonal tableaux

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