Vanishing Ideals of Lattice Diagram Determinants

A lattice diagram is a finite set L={(p1,q1),…,(pn,qn)} of lattice cells in the positive quadrant. The corresponding lattice diagram determinant is ΔL(Xn;Yn)=det ∥ xIpjyIqj∥. The space ML is the space spanned by all partial derivatives of ΔL(Xn;Yn). We denote by ML0 the Y-free component of ML. 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 Mμ0 and we give the first known description of the vanishing ideal of Mμ/ij0.