An extension of Viennot's shadow to rook placements via orbit harmonics
Jasper M. Liu, Hai Zhu
TL;DR
This paper extends Viennot's shadow framework to rook placements on an $n\times m$ board by introducing the ideal $I_{n,m,r}$ whose associated graded matches the orbit-harmonics quotient of the rook-locus ${\mathcal{UZ}}_{n,m,r}$. It constructs an extended-shadow monomial basis $\{\mathfrak{es}(\mathcal{R})\}$ indexed by rook placements and proves it is the standard monomial basis with respect to a diagonal monomial order, while also describing the graded $\mathfrak{S}_n\times\mathfrak{S}_m$-module structure via a refined Hilbert series $\mathrm{Hilb}(q)$ and graded Frobenius image. The key contributions include an explicit generating set for $\mathrm{gr}\,\mathbf{I}(\mathcal{UZ}_{n,m,r})$, a demonstration that $I_{n,m,r}=\mathrm{gr}\,\mathbf{I}(\mathcal{UZ}_{n,m,r})$, and a detailed description of the $\mathfrak{S}_n\times\mathfrak{S}_m$-module decomposition in terms of Schur-positivity and Lyndon-like shadow data. The results provide a concrete algebraic-combinatorial bridge between rook placements, shadow constructions, and symmetric-function representations with potential applications to log-concavity phenomena and colored generalizations.
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 $\mathbb{C}[\mathbf{x}_{n \times m}]/I_{n,m,r}$.