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

Abstract

For fixed positive integers $n,m,r$, let $\mathrm{Mat}_{n \times m}(\mathbb{C})$ 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 $\mathbb{C}[\mathbf{x}_{n \times m}]/I_{n,m,r}$ 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 $\mathbb{C}[\mathbf{x}_{n \times m}]/I_{n,m,r}$ with respect to diagonal monomial orders. We also determine the graded $\mathfrak{S}_n\times\mathfrak{S}_m$-module structure of $\mathbb{C}[\mathbf{x}_{n \times m}]/I_{n,m,r}$.