On Manin-Schechtman orders related to directed graphs

TL;DR

This paper generalizes Manin-Schechtman orders by introducing convex orders on path systems in directed graphs, establishing a poset structure with unique minimal and maximal elements, extending classical higher Bruhat order results.

Contribution

It extends the classical theory of higher Bruhat orders to a broader setting involving convex orders on directed graph path systems, with new local transformations and poset properties.

Findings

The set of convex orders forms a poset with unique minimal and maximal elements.

Local transformations, or flips, preserve the poset structure.

The generalization encompasses classical higher Bruhat orders as a special case.

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 $[n]=\{1,2,\ldots,n\}$. 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.