Lattice diagram polynomials and extended Pieri Rules

The lattice cell in the i + 1 st row and j + ist column of the positive qua drant of the plane is denoted (i,j). If mu is a partition of n + i, we deno te by mu/ij the diagram obtained by removing the cell (i, i) from the (Fren ch) Ferrers diagram of mu. We set Delta(mu/ij)= det \\x(i)(p)j y(i)(q)j\\(n )(i),(j=1) where (p(1), q(1)),..., (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 dia gonal action of S-n gives M-mu/ij the structure of a bigraded S-n-module. W e conjecture that M-mu/ij is always a direct sum of k left regular represen tations of S-n, where k is the number of cells that are weakly north and ea st of (i,i) in mu. We also make a number of conjectures describing the prec ise 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 re presentation theoretical interpretation of the coefficients appearing in so me Macdonald Pieri Rules. (C) 1999 Academic Press.