Rook placements and orbit harmonics

TL;DR

This paper applies orbit harmonics to analyze the structure of matrix loci defined by rook placements, providing explicit character formulas and demonstrating applications like module presentations and isomorphisms.

Contribution

It introduces new graded character formulas for modules associated with rook placement loci, extending orbit harmonics techniques to these combinatorial geometric objects.

Findings

Derived explicit signed and sign-free character formulas.

Established module injections and isomorphisms.

Provided concise presentations of the modules.

Abstract

For fixed positive integers $n,m$, let $\mathrm{Mat}_{n\times m}(\mathbb{C})$ be the affine space consisting of all $n\times m$ complex matrices, and let $\mathbb{C}[\mathbf{x}_{n\times m}]$ be its coordinate ring. For $0\le r\le\min\{m,n\}$, we apply the orbit harmonics method to the finite matrix loci $\mathcal{Z}_{n,m,r}$ of rook placements with exactly $r$ rooks, yielding a graded $\mathfrak{S}_n\times\mathfrak{S}_m$-module $R(\mathcal{Z}_{n,m,r})$. We find one signed and two sign-free graded character formulae for $R(\mathcal{Z}_{n,m,r})$. We also exhibit some applications of these formulae, such as proving a concise presentation of $R(\mathcal{Z}_{n,m,r})$, and proving some module injections and isomorphisms. Some of our techniques are still valid for involution matrix loci.