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 \(\mu\) 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 \(\Delta_{\mu/ij}=\det \| x_i^{p_j}y_i^{q_j} \|_{i,j=1}^n\), where \((p_1,q_1),... ,(p_n,q_n)\) are the cells of \(\mu/ij\), and let \({\bf M}_{\mu/ij}\) 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 \({\bf M}_{\mu/ij}\) the structure of a bigraded \(S_n\)-module. We conjecture that \({\bf M}_{\mu/ij}\) 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 \({\bf M}_{\mu/ij}\) 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.