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Séminaire Lotharingien de Combinatoire, 80B.70 (2018), 12 pp.

# Carolina Benedetti, Nantel Bergeron, and John Machacek

# Hypergraphic Polytopes: Combinatorial Properties and Antipode

**Abstract.**
Given a hypergraph *G*, its hypergraphic polytope *P*_{G} is the Minkowski sum of simplices corresponding to each hyperedge of *G*.
Using a notion of orientation on *G*, we prove that the faces of *P*_{G} are in bijective correspondence with acyclic orientations of *G*.
This allows us to give a geometric understanding of the antipode in a cocommutative Hopf algebra of hypergraphs.
We also give a characterization of when a hypergraphic polytope is a simple polytope.
The correspondence between faces and acyclic orientations is used to
prove some combinatorial properties of nestohedra and generalized
Pitman-Stanley polytopes.

Received: November 14, 2017.
Accepted: February 17, 2018.
Final version: April 1, 2018.

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