Rook placements and orbit harmonics
Hai Zhu
TL;DR
This work develops an orbit-harmonics approach to finite rook-placement loci ${\mathcal Z}_{n,m,r}$, producing a graded ${\mathfrak S}_n\times{\mathfrak S}_m$-module $R({\mathcal Z}_{n,m,r})$ and two families of graded Frobenius formulas. The authors first realize $R({\mathcal Z}_{n,m,r})$ as a quotient by a defining ideal ${I_{n,m}^{(r)}}$ and prove its equality with the associated graded vanishing ideal, enabling a monomial-basis description via rook monomials. A signed graded character is derived as ${\mathrm{grFrob}}(R({\mathcal Z}_{n,m,r});q)=\sum_{d=0}^r q^d\cdot{\{SF_d-SF_{d-1}\}}_{\lambda_1\le n+m-d-r}$, and a positive, sign-free refinement is produced through intricate lattice-path bijections and shadow maps, yielding a combinatorial expression in terms of HS and PHS sets. The results lead to concise presentations, module injections, and connections to involution-matrix loci and upper rook loci, offering tools toward monotonicity and log-concavity questions in equivariant settings and suggesting several open problems, including generalizations to colored rook placements.
Abstract
For fixed positive integers $n,m$, let $\mathrm{Mat}_{n\times m}(\mathbb{C})$ be the affine space consisting of all $n\times m$ complex matrices, and let $\mathbb{C}[\mathbf{x}_{n\times m}]$ be its coordinate ring. For $0\le r\le\min\{m,n\}$, we apply the orbit harmonics method to the finite matrix loci $\mathcal{Z}_{n,m,r}$ of rook placements with exactly $r$ rooks, yielding a graded $\mathfrak{S}_n\times\mathfrak{S}_m$-module $R(\mathcal{Z}_{n,m,r})$. We find one signed and two sign-free graded character formulae for $R(\mathcal{Z}_{n,m,r})$. We also exhibit some applications of these formulae, such as proving a concise presentation of $R(\mathcal{Z}_{n,m,r})$, and proving some module injections and isomorphisms. Some of our techniques are still valid for involution matrix loci.