On Manin-Schechtman orders related to directed graphs

Abstract

As a generalization of weak Bruhat orders on permutations, in 1989 Manin and Schechtman introduced the notion of a higher Bruhat order on the $d$-element subsets of a set $[n]=\{1,2,\dots ,n\}$. Among other results in this field, they proved that the set of such orders for $n,d$ fixed, endowed with natural local transformations, constitutes a poset with one minimal and one maximal elements. In this paper we consider a wider model, involving the so-called convex order on certain path systems in an acyclic directed graph, introduce local transformations, or flips, on such orders and prove that the resulting structure gives a poset with one minimal and one maximal elements as well, yielding a generalization of the above-mentioned classical result.

Theorem 2.1 is not valid for some acyclic directed graphs