RELATIONS BETWEEN THE CLAUSEN AND KAZHDAN-LUSZTIG REPRESENTATIONS OF Sn

We use Kazhdan-Lusztig polynomials and subspaces of the polynomial ring C[x1,1, . . . , xn,n] to construct irreducible Sn-modules. This construction produces exactly the same matrices as the Kazhdan-Lusztig construction [Invent.Math 53 (1979)], but does not employ the Kazhdan-Lusztig preorders. It also produces exactly the same modules as those which Clausen constructed using a different basis in [J. Symbolic Comput. 11 (1991)]. We show that the two resulting matrix representations are related by a unitriangular transition matrix. This provides a C[x1,1, . . . , xn,n]-analog of results due to Garsia and McLarnan [Adv.Math. 69 (1988)], and McDonough and Pallikaros [J.Pure Appl. Alg. 203 (2005)].