Matrix Schubert varieties and Gaussian conditional independence models

摘要

Matrix Schubert varieties are certain varieties in the affine space of squarematrices which are determined by specifying rank conditions on submatrices. Westudy these varieties for generic matrices, symmetric matrices, and uppertriangular matrices in view of two applications to algebraic statistics: weobserve that special conditional independence models for Gaussian randomvariables are intersections of matrix Schubert varieties in the symmetric case.Consequently, we obtain a combinatorial primary decomposition algorithm forsome conditional independence ideals. We also characterize the vanishing idealsof Gaussian graphical models for generalized Markov chains. In the course of this investigation, we are led to consider three relatedstratifications, which come from the Schubert stratification of a flag variety.We provide some combinatorial results, including describing the stratificationsusing the language of rank arrays and enumerating the strata in each case.