Hypergraphic polytopes: combinatorial properties and antipode

TL;DR

This paper explores the combinatorial and geometric properties of hypergraphic polytopes, establishing a bijection between hypergraph orientations and polytope faces, and providing new insights into the antipode map in hypergraph Hopf algebras.

Contribution

It introduces a new geometric interpretation of the antipode coefficients and characterizes hypergraphs that produce simple hypergraphic polytopes, extending known results to new classes.

Findings

Bijection between acyclic hypergraph orientations and hypergraphic polytope faces

Geometric interpretation of antipode coefficients in hypergraph Hopf algebra

Characterization of hypergraphs leading to simple hypergraphic polytopes

Abstract

In an earlier paper, the first two authors defined orientations on hypergraphs. Using this definition we provide an explicit bijection between acyclic orientations in hypergraphs and faces of hypergraphic polytopes. This allows us to obtain a geometric interpretation of the coefficients of the antipode map in a Hopf algebra of hypergraphs. This interpretation differs from similar ones for a different Hopf structure on hypergraphs provided recently by Aguiar and Ardila. Furthermore, making use of the tools and definitions developed here regarding orientations of hypergraphs we provide a characterization of hypergraphs giving rise to simple hypergraphic polytopes in terms of acyclic orientations of the hypergraph. In particular, we recover this fact for the nestohedra and the hyper-permutahedra, and prove it for generalized Pitman-Stanley polytopes as defined here.