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

Abstract: For fixed positive integers $n,m,r$ with $r \leq \min(n,m)$, 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 $\mathbb{C}[\mathbf{x}{n \times n}]/I{n,m,r}$ carries an action of the group $\mathfrak{S}n \times \mathfrak{S}_m$ by independent permutations of rows and columns. This quotient ring is obtained by applying the orbit harmonics method to matrix loci corresponding to all rook placements of size at least $r$. We show that the quotient $\mathbb{C}[\mathbf{x}{n \times n}]/I_{n,m,r}$ admits a standard monomial basis, which is determined by extending rook placements to an element in $\mathfrak{S}{n+m-r}$ and applying Viennot's shadow line avatar of the Schensted correspondence. We also determine the graded $\mathfrak{S}_n\times\mathfrak{S}_m$-module structure of $\mathbb{C}[\mathbf{x}{n \times n}]/I_{n,m,r}$.