Zigzags, contingency tables, and quotient rings
Jaeseong Oh, Brendon Rhoades
TL;DR
The paper defines the homogeneous ideal $I_{\alpha,\beta}$ generated by row sums, column sums, and degree-restriction monomials, and studies the quotient ring $R_{\alpha,\beta}=\mathbb{F}[{\mathbf{x}}_{k\times p}]/I_{\alpha,\beta}$. It proves the Hilbert series satisfies $\mathrm{Hilb}(R_{\alpha,\beta};q)=\sum_{A\in\mathcal{C}_{\alpha,\beta}} q^{n-\operatorname{zigzag}(A)}$ and identifies a standard monomial basis given by the matrix-ball avatar of the RSK correspondence. The quotient carries a graded action of $\mathrm{Stab}(\alpha)\times\mathrm{Stab}(\beta)$, with an ungraded isomorphism $R_{\alpha,\beta} \cong \mathbb{F}[\mathcal{C}_{\alpha,\beta}]$ and a description of its graded refinement; the paper develops orbit-harmonics techniques to relate the contingency-table locus to the quotient ring. It introduces auxiliary one-row quotient rings and polarization methods to control leading terms and extend diagonal information to the full matrix, situating the results in a matrix-ball/RSK framework. Overall, the work extends previous permutation-matrix results to general contingency tables, revealing a deep link between zigzag statistics, matrix-ball geometry, and graded representation theory via orbit harmonics.
Abstract
Let $\mathbf{x}_{k \times p}$ be a $k \times p$ matrix of variables and let $\mathbb{F}[\mathbf{x}_{k \times p}]$ be the polynomial ring in these variables. Given two weak compositions $α,β\models_0 n$ of lengths $\ell(α) = k$ and $\ell(β) = p$, we study the ideal $I_{α,β} \subseteq \mathbb{F}[\mathbf{x}_{k \times \ell}]$ generated by row sums, column sums, monomials in row $i$ of degree $> α_i$, and monomials in column $j$ of degree $> β_j$. We prove results connecting algebraic properties of the quotient ring $R_{α,β} := \mathbb{F}[\mathbf{x}_{k \times \ell}]/I_{α,β}$ with the set $C_{α,β}$ of $α,β$-contingency tables. The standard monomial basis of $R_{α,β}$ with respect to a diagonal term order is encoded by the matrix-ball avatar of the RSK correspondence. We describe the Hilbert series of $R_{α,β}$ in terms of a zigzag statistic on contingency tables. The ring $R_{α,β}$ carries a graded action of the product $\mathrm{Stab}(α) \times \mathrm{Stab}(β)$ of symmetry groups of the sequences $α= (α_1,\dots,α_k)$ and $β= (β_1,\dots,β_p)$; we describe how to calculate the isomorphism type of this graded action. Our analysis regards the set $C_{α,β}$ as a locus in the affine space $\mathrm{Mat}_{k \times p}(\mathbb{F})$ and applies orbit harmonics to this locus.