Plabic graphs and zonotopal tilings

TL;DR

This paper establishes a connection between maximal collections of chord separated sets, plabic graphs, and zonotopal tilings, revealing their structural relationships and geometric interpretations.

Contribution

It introduces a new characterization of maximal chord separated sets as vertex labels of zonotopal tilings, linking combinatorics, geometry, and graph theory.

Findings

Maximal chord separated sets are also maximal by size.

Such collections correspond to vertex labels of fine zonotopal tilings.

Plabic graphs and square moves naturally arise in the tiling framework.

Abstract

We say that two sets $S,T\subset\{1,2,\dots,n\}$ are chord separated if there does not exist a cyclically ordered quadruple $a,b,c,d$ of integers satisfying $a,c\in S-T$ and $b,d\in T-S$. This is a weaker version of Leclerc and Zelevinsky's weak separation. We show that every maximal by inclusion collection of pairwise chord separated sets is also maximal by size. Moreover, we prove that such collections are precisely vertex label collections of fine zonotopal tilings of the three-dimensional cyclic zonotope. In our construction, plabic graphs and square moves appear naturally as horizontal sections of zonotopal tilings and their mutations respectively.