Séminaire Lotharingien de Combinatoire, 91B.21 (2024), 12 pp.
Jonah Berggren and Khrystyna Serhiyenko
Wilting Theory of Flow Polytopes
Abstract.
Many important polytopes and their canonical triangulations appear as
DKK triangulations of a framed directed acyclic graph (DAG)
Γ. These triangulations are combinatorially modelled by cliques
of routes on the framed DAG. When Γ is amply framed, the dual
graph of its DKK triangulation, or DKK graph, has a lattice structure
called the DKK lattice.
We study the clique complex of routes which avoid an arbitrary
set of "wilted" edges. This leads to various decompositions of the
DKK lattice into intervals, generalizing decompositions of the Tamari
lattice into ν-Tamari intervals. We further classify the framed
DAGs whose DKK graphs may be understood as an interval in the DKK
lattice of an amply framed DAG.
We realize ν-Tamari lattices and the s-weak
order as DKK lattices of such "rooted" DAGs and we extend results
about shellability and h*-polynomials from the amply framed case to
the rooted case.
Received: November 15, 2023.
Accepted: February 15, 2024.
Final version: April 1, 2024.
The following versions are available: