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

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.