Zigzags, contingency tables, and quotient rings

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 $\alpha,\beta \models_0 n$ of lengths $\ell(\alpha) = k$ and $\ell(\beta) = p$, we study the ideal $I_{\alpha,\beta} \subseteq \mathbb{F}[\mathbf{x}_{k \times \ell}]$ generated by row sums, column sums, monomials in row $i$ of degree $> \alpha_i$, and monomials in column $j$ of degree $> \beta_j$. We prove results connecting algebraic properties of the quotient ring $R_{\alpha,\beta} := \mathbb{F}[\mathbf{x}_{k \times \ell}]/I_{\alpha,\beta}$ with the set $C_{\alpha,\beta}$ of $\alpha,\beta$-contingency tables. The standard monomial basis of $R_{\alpha,\beta}$ with respect to a diagonal term order is encoded by the matrix-ball avatar of the RSK correspondence. We describe the Hilbert series of $R_{\alpha,\beta}$ in terms of a zigzag statistic on contingency tables. The ring $R_{\alpha,\beta}$ carries a graded action of the product $\mathrm{Stab}(\alpha) \times \mathrm{Stab}(\beta)$ of symmetry groups of the sequences $\alpha = (\alpha_1,\dots,\alpha_k)$ and $\beta = (\beta_1,\dots,\beta_p)$; we describe how to calculate the isomorphism type of this graded action. Our analysis regards the set $C_{\alpha,\beta}$ as a locus in the affine space $\mathrm{Mat}_{k \times p}(\mathbb{F})$ and applies orbit harmonics to this locus.