Contingency Tables with Variable Margins (with an Appendix by Pavel Etingof)

TL;DR

This paper explores the combinatorics of cell decompositions of symmetric products of the complex line, linking contingency matrices to stratifications, sheaf constructibility, and polyhedral structures, with applications to algebraic and topological contexts.

Contribution

It introduces a new criterion for sheaf constructibility related to contingency decompositions and studies the properties of the stochastihedron polyhedron, connecting combinatorics, topology, and algebraic geometry.

Findings

Contingency matrices parametrize cells in a quasi-regular decomposition.

A criterion is established for sheaves to be constructible with respect to different stratifications.

The meta-matrix of contingency matrices is shown to be totally positive.

Abstract

Motivated by applications to perverse sheaves, we study combinatorics of two cell decompositions of the symmetric product of the complex line, refining the complex stratification by multiplicities. Contingency matrices, appearing in classical statistics, parametrize the cells of one such decomposition, which has the property of being quasi-regular. The other, more economical, decomposition, goes back to the work of Fox-Neuwirth and Fuchs on the cohomology of braid groups. We give a criterion for a sheaf constructible with respect to the ''contingency decomposition'' to be constructible with respect to the complex stratification. We also study a polyhedral ball which we call the stochastihedron and whose boundary is dual to the two-sided Coxeter complex (for the root system $A_n$) introduced by T.K. Petersen. The Appendix by P. Etingof studies enumerative aspects of contingency matrices. In particular, it is proved that the ''meta-matrix'' formed by the numbers of contingency matrices of various sizes, is totally positive.