On certain spaces of lattice diagram polynomials

TL;DR

This paper investigates the structure and dimension bounds of certain lattice diagram polynomial spaces, generalizing the $n!$ conjecture and constructing explicit bases using symmetric differential operators.

Contribution

It introduces a new class of polynomial spaces related to lattice diagrams, provides upper bounds for their dimensions, and constructs explicit bases in specific cases.

Findings

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

Conjectured the optimality of the dimension bound

Constructed explicit bases for the subspace $M^k_{i,j}(X)$

Abstract

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.