Séminaire Lotharingien de Combinatoire, 80B.68 (2018), 12 pp.
Plabic Graphs and Zonotopal Tilings
We introduce the notion of chord separation of two sets which
generalizes 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 in bijection with fine zonotopal tilings of the
three-dimensional cyclic zonotope. As a result, we get that
Postnikov's reduced plabic graphs are precisely the objects dual to
horizontal sections of zonotopal tilings of the three-dimensional
cyclic zonotope, and Postnikov's moves on plabic graphs correspond to
flips of these zonotopal tilings.
Received: November 14, 2017.
Accepted: February 17, 2018.
Final version: April 1, 2018.
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