On certain spaces of lattice diagram determinants

TL;DR

This paper investigates the structure and dimensions of certain lattice diagram polynomial spaces, providing upper bounds and explicit bases for specific subspaces related to Ferrers diagrams and their cell removals.

Contribution

It introduces new upper bounds for the dimensions of lattice diagram polynomial spaces after cell removal and constructs explicit bases for particular subspaces.

Findings

Derived an upper bound for the dimension of $M^k_{i,j}(X,Y)$

Conjectured the optimality of the upper bounds

Constructed explicit bases for $M^k_{i,j}(X)$ with zero $Y$-degree

Abstract

The aim of this work is to study some lattice diagram polynomials $\Delta_D(X,Y)$. We recall that $M_D$ denotes the space of all partial derivatives of $\Delta_D$. In this paper, we want to study the space $M^k_{i,j}(X,Y)$ which is the sum of $M_D$ spaces where the lattice diagrams $D$ are obtained by removing $k$ cells from a given partition, these cells being in the ``shadow'' of a given cell $(i,j)$ of the 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. These upper bounds allow us to construct explicit bases for the subspace $M^k_{i,j}(X)$ consisting of elements of 0 $Y$-degree.