Relations between the Clausen and Kazhdan-Lusztig representations of the symmetric group

We use Kazhdan-Lusztig polynomials and subspaces of the polynomial ring C[x_1,1,…,x_n,n] to give a new construction of the Kazhdan-Lusztig representations of S_n. This construction produces exactly the same modules as those which Clausen constructed using a different basis in [M. Clausen, Multivariate polynomials, standard tableaux, and representations of symmetric groups, J. Symbolic Comput. (11), 5-6 (1991) 483-522. Invariant-theoretic algorithms in geometry (Minneapolis, MN, 1987)], and does not employ the Kazhdan-Lusztig preorders. We show that the two resulting matrix representations are related by a unitriangular transition matrix. This provides a C[x_1,1,…,x_n,n]-analog of results due to Garsia and McLarnan, and McDonough and Pallikaros, who related the Kazhdan-Lusztig representations to Young’s natural representations.