Rook placements and orbit harmonics

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.