Zigzags, contingency tables, and quotient rings

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.