Vanishing ideals of Lattice Diagram determinants

Abstract

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

15 pages, 3 figures (LaTeX2e with epsfig). A nice description of ideals for lattice diagrams

References in corpus (1)

• Lattice Diagram polynomials in one set of variables

Cited by in corpus (5)

• Catalan paths, Quasi-symmetric functions and Super-Harmonic Spaces
• Lattice Diagram polynomials in one set of variables
• Resolutions of De Concini-Procesi ideals indexed by hooks
• Schur Partial Derivative Operators
• On certain spaces of lattice diagram polynomials