Multiple Left Regular Representations Generated

Let p 1 p n 0, and p = detkx p j i k n i;j=1. Let Mp be the linear span of the partial derivatives of p. Then Mp is a graded Sn-module. We prove that it is the direct sum of graded left regular representations of Sn. Speciically, set j = p j , n , j, and let t be the Hilbert polynomial of the span of all skew Schur functions s = as varies in. Then the graded Frobenius characteristic of Mp is t ~ H 1 nx; q;t , a m ultiple of a Macdon-ald polynomial. Corresponding results are also given for the span of partial derivatives of an alternant o v er any complex reeection group. Let i;j denote the lattice cell in the i + 1 st row and j + 1 st column of the positive quadrant of the plane. If L is a diagram with lattice cells p 1 ; q 1 ; : : : ; p n ; q n , we set L = detkx p j i y q j i k n i;j=1 , and let M L be the linear span of the partial derivatives of L. The bihomogeneity o f L and its alternating nature under the diagonal action of Sn gives M L the structure of a bigraded Sn-module. We give a family of examples and some general conjectures about the bivariate Frobenius characteristic of M L for two dimensional diagrams. * Work carried out under NSERC and FCAR grant support. y Work carried out under NSF grant support.