Lattice Diagram Polynomials and Extended Pieri Rules

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 is a partition of n + 1 , w e denote by =ij the diagram obtained by removing the cell i; j from the French Ferrers diagram of. We set =ij = det k x pj i y qj be the linear span of the partial derivatives of =ij. The bihomogeneity o f =ij and its alternating nature under the diagonal action of S n gives M =ij the structure of a bigraded S n-module. We conjecture that M =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 i n. W e also make a n umber of conjectures describing the precise nature of the bivariate Frobenius characteristic of M =ij in terms of the theory of Macdonald polynomials. On the validity of these conjectures, we derive a n umber of surprising identities. In particular, we obtain a representation theoretical interpretation of the coeecients appearing in some Macdonald Pieri Rules.