Spaces of Lattice Diagram Polynomials in One Set of Variables

Abstract

For a partition μ of n, let Mμ be the space span of all partial derivatives of the alternate associated to μ in two n-sets of variables X and Y. The n!-theorem of [M. Haiman, J. Amer. Math. Soc.14 (2001), 941–1006] states that the dimension of Mμ is n!. In [F. Bergeron, N. Bergeron, A. Garsia, M. Haiman, and G. Tesler, Adv. Math.142 (1999), 244–334], we introduced slightly more general spaces Mμ/ij as a tool for an elementary proof of the n!-theorem. For this one needs a formula for the bigraded characters in terms of Macdonald polynomials that would follow from a representation theoretic proof of our four-term recurrence. Here we give an explicit basis for the Y-free component of the space Mμ/ij, and using this description we prove the Y-free analog of the four-term recurrence. Our basis of Mμ/ij has the further nice feature that it is a natural generalization of the Artin basis for the space of harmonic (covariant) polynomials for the symmetric group.