Abstract Our main result here is that, under a suitable order of standard tableaux, the classical representation of Sn introduced by Young (in “The Collected Papers of Alfred Young, 1873–1940,” Univ. of Toronto Press, Toronto) (QSA IV), and usually referred as the Natural representattion, the the more recently discovered (Invent. Math.53 (1979), 165–184) Kazhdan-Lusztig (K-L) representation are related by an upper triangular integral matrix with unit diagonal elements. We have been led to this discovery by a numerical exploration. We noted it in each of the irreducible representations of Sn up to n = 6. The calculations in these cases were carried out by constructing the corresponding Kazhdan-Lusztig graphs from tables (M. Goresky, Tables of Kazhdan-Lusztig polynomials, unpublished) of K-L polynomials. To extend the calculations to n = 7 we have used graphs obtained by means of an algorithm given by Lascoux and Schutzenberger (Polynomes de Kazhdan & Lusztig pour les Grassmanniennes, preprint). Remarkably, the same property holds also for these graphs. These findings appear to confirm the assertion made by these authors that their algorithm does indeed yield K-L graphs. For the case of hook shapes we have obtained an explicit construction of the transforming matrices, a result which was also suggested by our numerical data. For general shapes, the transforming matrices are less explicit and our proof is based on certain properties of the Kazhdan-Lusztig representations given in their article (Invent. Math.53 (1979), 165–184) and on a purely combinatorial construction of the natural representation.
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