Lattice Diagram Polynomials and Extended Pieri Rules

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F. Bergeron, N. Bergeron, A. Garsia, M. Haiman, G. Tesler,
Lattice Diagram Polynomials and Extended Pieri Rules,
Advances in Mathematics,142 no. 2 (1999), 244-334.

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Abstract

The lattice cell in the i+1^st row and j+1^st column of the positive quadrant of the plane is denoted (i,j). If \mu is a partition of n+1, we denote by \mu/ij the diagram obtained by removing the cell (i,j) from the (French) Ferrers diagram of \mu. We set \Delta_\mu/ij=det[x_i ^p_j y_i ^q_j]ⁿ_i,j=1, where (p₁,q₁),...,(p_n,q_n) are the cells of \mu/ij, and let M_\mu/ij be the linear span of the partial derivatives of \Delta_\mu/ij. The bihomogeneity of \Delta_\mu/ij and its alternating nature under the diagonal action of S_n gives M_\mu/ij the structure of a bigraded S_n-module. We conjecture that M_\mu/ij is always a direct sum of k left regular representations of S_n, where k is the number of cells that are weakly north and east of (i,j) in \mu. We also make a number of conjectures describing the precise nature of the bivariate Frobenius characteristic of M_\mu/ij in terms of the theory of Macdonald polynomials. On the validity of these conjectures, we derive a number of surprising identities. In particular, we obtain a representation theoretical interpretation of the coefficients appearing in some Macdonald Pieri Rules.