Brewing Fubini-Bruhat Orders

Abstract: The Bruhat order on permutations arises out of the study of Schubert varieties in Grassmannians and flag varieties, which have been important for over 100 years. The purpose of this paper is to study variations on this theme related to subvarieties of the spanning line configurations $X_{n,k}$ as defined by Pawlowski and Rhoades. These subvarieties are indexed by Fubini words, or equivalently by ordered set partitions. Three natural partial orders arise in this context; we refer to them as the decaf, medium roast, and espresso orders. The decaf order is a generalization of the weak order on permutations defined by covering relations using simple transpositions. The medium roast order is a generalization of the (strong) Bruhat order defined by the closure relationship on the subvarieties. The espresso order is the transitive closure of a relation based on intersecting subvarieties. Many properties of Schubert varieties and Bruhat order extend to one or more of the three Fubini-Bruhat orders. We examine some of the many possibilities in this work.