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Vanishing ideals of Lattice Diagram determinants

A lattice diagram is a finite set $L=\{(p_1,q_1),... ,(p_n,q_n)\}$ of lattice cells in the positive quadrant. The corresponding lattice diagram determinant is $Δ_L(\X;\Y)=\det \| x_i^{p_j}y_i^{q_j} \|$. The space $M_L$ is the space spanned by all partial derivatives of $Δ_L(\X;\Y)$. We denote by $M_L^0$ the $Y$-free component of $M_L$. For $μ$ a partition of $n+1$, we denote by $μ/ij$ the diagram obtained by removing the cell $(i,j)$ from the Ferrers diagram of $μ$. Using homogeneous partially symmetric polynomials, we give here a dual description of the vanishing ideal of the space $M_μ^0$ and we give the first known description of the vanishing ideal of $M_{μ/ij}^0$.