Multiple Left Regular Representations Generated by Alternants

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F. Bergeron, A.M. Garsia, G. Tesler,
Multiple Left Regular Representations Generated by Alternants,
Journal of Combinatorial Theory, Series A, to appear.

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Abstract

Let p₁>...>p_n>=0, and Delta_p=det| x_i^p_j |_i,j=1ⁿ. Let M_p be the linear span of the partial derivatives of Delta_p. Then M_p is a graded S_n-module. We prove that it is the direct sum of graded left regular representations of S_n. Specifically, set lambda_j=p_j-(n-j), and let Xi_lambda(t) be the Hilbert polynomial of the span of all skew Schur functions s_{lambda / mu} as mu varies in lambda. Then the graded Frobenius characteristic of M_p is Xi_lambda(t) H^~_1ⁿ(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+1}^st row and {j+1}^st column of the positive quadrant of the plane. If L is a diagram with lattice cells (p₁,q₁),...,(p_n,q_n), we set Delta_L=det| x_i^p_j y_i^q_j |_i,j=1ⁿ, and let M_L be the linear span of the partial derivatives of Delta_L. The bihomogeneity of Delta_L and its alternating nature under the diagonal action of S_n gives M_L the structure of a bigraded S_n-module. We give a family of examples and some general conjectures about the bivariate Frobenius characteristic of M_L for two dimensional diagrams.