Lattice Diagram polynomials in one set of variables

The space $M_{\mu/i,j}$ spanned by all partial derivatives of the lattice polynomial $\Delta_{\mu/i,j}(X;Y)$ is investigated in math.CO/9809126 and many conjectures are given. Here, we prove all these conjectures for the $Y$-free component $M_{\mu/i,j}^0$ of $M_{\mu/i,j}$. In particular, we give an explicit bases for $M_{\mu/i,j}^0$ which allow us to prove directly the central {\sl four term recurrence} for these spaces.Comment: 15 page

On certain spaces of lattice diagram polynomials

The aim of this work is to study some lattice diagram determinants $\Delta_L(X,Y)$. We recall that $M_L$ denotes the space of all partial derivatives of $\Delta_L$. In this paper, we want to study the space $M^k_{i,j}(X,Y)$ which is defined as the sum of $M_L$ spaces where the lattice diagrams $L$ are obtained by removing $k$ cells from a given partition, these cells being in the ``shadow'' of a given cell $(i,j)$ in a fixed Ferrers diagram. We obtain an upper bound for the dimension of the resulting space $M^k_{i,j}(X,Y)$, that we conjecture to be optimal. This dimension is a multiple of $n!$ and thus we obtain a generalization of the $n!$ conjecture. Moreover, these upper bounds associated to nice properties of some special symmetric differential operators (the ``shift'' operators) allow us to construct explicit bases in the case of one set of variables, i.e. for the subspace $M^k_{i,j}(X)$ consisting of elements of 0 $Y$-degree

Resolutions of De Concini-Procesi ideals of hooks

We find a minimal generating set for the defining ideal of the schematic intersection of the set of diagonal matrices with the closure of the conjugacy class of a nilpotent matrix indexed by a hook partition. The structure of this ideal allows us to compute its minimal free resolution and give an explicit description of the graded Betti numbers, and study its Hilbert series and regularity

The defining ideals of conjugacy classes of nilpotent matrices and a conjecture of Weyman

Tanisaki introduced generating sets for the defining ideals of the schematic intersections of the closure of conjugacy classes of nilpotent matrices with the set of diagonal matrices. These ideals are naturally labeled by integer partitions. Given such a partition λ, we define several methods to produce a reduced generating set for the associated ideal Iλ. For particular shapes we find nice generating sets. By comparing our sets with some generating sets of Iλ arising from a work of Weyman, we find a counterexample to a related conjecture of Weyman.Natural Sciences and Engineering Research Council of CanadaMinisterio de Educación y Cienci

Vanishing ideals of Lattice Diagram determinants

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

Vanishing ideals of Lattice Diagram determinants

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