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 ∆L(Xn;Yn) = det ‖ x pj i y qj i ‖. The space ML is the space spanned by all partial derivatives of ∆L(Xn;Yn). We denote by M 0 L 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 M μ and we give the first known description of the vanishing ideal of M μ/ij .