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 $\hat{I}\pm ,{\hat{I}}^2{\models }_{0}n$ of lengths $\ell (\hat{I}\pm )=k$ and $\ell ({\hat{I}}^2)=p$, we study the ideal $I_{\hat{I}\pm ,{\hat{I}}^2}\subseteq \mathbb{F}[{\mathbf{x}}_{k\times \ell }]$ generated by row sums, column sums, monomials in row $i$ of degree $>\hat{I}{\pm }_i$, and monomials in column $j$ of degree $>{\hat{I}}_j^2$. We prove results connecting algebraic properties of the quotient ring $R_{\hat{I}\pm ,{\hat{I}}^2}:=\mathbb{F}[{\mathbf{x}}_{k\times \ell }]/I_{\hat{I}\pm ,{\hat{I}}^2}$ with the set $C_{\hat{I}\pm ,{\hat{I}}^2}$ of $\hat{I}\pm ,{\hat{I}}^2$-contingency tables. The standard monomial basis of $R_{\hat{I}\pm ,{\hat{I}}^2}$ with respect to a diagonal term order is encoded by the matrix-ball avatar of the RSK correspondence. We describe the Hilbert series of $R_{\hat{I}\pm ,{\hat{I}}^2}$ in terms of a zigzag statistic on contingency tables. The ring $R_{\hat{I}\pm ,{\hat{I}}^2}$ carries a graded action of the product $\mathrm{Stab}(\hat{I}\pm )\times \mathrm{Stab}({\hat{I}}^2)$ of symmetry groups of the sequences $\hat{I}\pm =(\hat{I}{\pm }_1,\dots ,\hat{I}{\pm }_k)$ and ${\hat{I}}^2=({\hat{I}}_1^2,\dots ,{\hat{I}}_p^2)$; we describe how to calculate the isomorphism type of this graded action. Our analysis regards the set $C_{\hat{I}\pm ,{\hat{I}}^2}$ as a locus in the affine space ${\mathrm{Mat}}_{k\times p}(\mathbb{F})$ and applies orbit harmonics to this locus.

38 pages, 1 figure