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 $μ$ is a partition of $n+1$, we denote by $μ/ij$ the diagram obtained by removing the cell $(i,j)$ from the (French) Ferrers diagram of $μ$. We set $Δ_{μ/ij}=\det \| x_i^{p_j}y_i^{q_j} \|_{i,j=1}^n$, where $(p_1,q_1),... ,(p_n,q_n)$ are the cells of $μ/ij$, and let ${\bf M}_{μ/ij}$ be the linear span of the partial derivatives of $Δ_{μ/ij}$. The bihomogeneity of $Δ_{μ/ij}$ and its alternating nature under the diagonal action of $S_n$ gives ${\bf M}_{μ/ij}$ the structure of a bigraded $S_n$-module. We conjecture that ${\bf M}_{μ/ij}$ 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 $μ$. We also make a number of conjectures describing the precise nature of the bivariate Frobenius characteristic of ${\bf M}_{μ/ij}$ 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.

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