## Vanishing ideals of Lattice Diagram determinants

J.-C.~Aval and N.~Bergeron
(Bordeaux and York)

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