Multiple Left Regular Representations Generated by Alternants

Abstract

Let p1>…>pn⩾0, and Δp=det‖xpji‖ni, j=1. Let Mp be the linear span of the partial derivatives of Δp. Then Mp is a graded Sn-module. We prove that it is the direct sum of graded left regular representations of Sn. Specifically, set λj=pj−(n−j), and let Ξλ(t) be the Hilbert polynomial of the span of all skew Schur functions sλ/μ as μ varies in λ. Then the graded Frobenius characteristic of Mp is Ξλ(t)H1n(x; q, t), a multiple of a Macdonald polynomial. Corresponding results are also given for the span of partial derivatives of an alternant over any complex reflection group. Let (i, j) denote the lattice cell in the i+1st row and j+1st column of the positive quadrant of the plane. If L is a diagram with lattice cells (p1, q1), …, (pn, qn), we set ΔL=det‖xpjiyqji‖ni, j=1, and let ML be the linear span of the partial derivatives of ΔL. The bihomogeneity of ΔL and its alternating nature under the diagonal action of Sn gives ML the structure of a bigraded Sn-module. We give a family of examples and some general conjectures about the bivariate Frobenius characteristic of ML for two dimensional diagrams.