An extension of Viennot's shadow to rook placements via orbit harmonics

For fixed positive integers $n,m,r$, let ${\mathrm{M}\mathrm{a}\mathrm{t}}_{n\times m}(\mathbb{C})$ be the affine space of $n\times m$ complex matrices with coordinate ring $\mathbb{C}[{\mathbf{x}}_{n\times m}]$. We define a homogeneous ideal $I_{n,m,r}$, where the graded quotient $\mathbb{C}[{\mathbf{x}}_{n\times m}]\mathrm{/}{I}_{n,m,r}$ is obtained from the orbit harmonics deformation of the matrix loci corresponding to all rook placements of size at least $r$. By extending rook placements to elements in ${\mathfrak{S}}_{n+m-r}$ and applying Viennot’s shadow line avatar of the Schensted correspondence, we compute the standard monomial basis of the quotient $\mathbb{C}[{\mathbf{x}}_{n\times m}]\mathrm{/}{I}_{n,m,r}$ with respect to diagonal monomial orders. We also determine the graded ${\mathfrak{S}}_n\times {\mathfrak{S}}_m$-module structure of $\mathbb{C}[{\mathbf{x}}_{n\times m}]\mathrm{/}{I}_{n,m,r}$.