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

TL;DR

This paper extends Viennot's shadow line method to rook placements using orbit harmonics, computing bases and module structures of related matrix loci ideals, advancing algebraic combinatorics and representation theory.

Contribution

It introduces a new approach to analyze matrix loci associated with rook placements via orbit harmonics and Viennot's shadow line technique, providing explicit bases and module structures.

Findings

Computed the standard monomial basis of the quotient algebra.

Determined the graded module structure under symmetric group actions.

Extended rook placements to elements in symmetric groups for combinatorial analysis.

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 $\mathbb{C}[\mathbf{x}_{n \times m}]/I_{n,m,r}$.