A bruhat order for bipartite graphs whose node sets are posets: Lifting, switching, and adding edges

Abstract

Consider two partially ordered setsP, Q and a number of edges connecting some of the points ofP with some of the points ofQ. This yields a bipartite graph. Some pairs of the edges may cross each other because their endpoints atP andQ are oppositely ordered. A natural decrossing operation is to exchange the endpoints of these edges incident atQ, say. This is called a switch. A left lift of an edge means to replace its starting point atP by a larger starting point. A right lift is defined symmetrically for the endpoints atQ. The operation of adding an edge cannot, informally, be explained better. Assume we are given two bipartite graphs π, σ on the node setPσQ. We show that for certain pairs (P, Q) of finite posets, a neat necessary and sufficient criterion can be given in order that σ is obtainable from π by the sequence of elementary operations just defined. A recent characterization of the Bruhat order of the symmetric group follows as a special case.