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

Abstract

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.