## 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

$M_{\mu/ij}^0$.