Hopf monoids and generalized permutahedra

TL;DR: The generalized permutahedra are a family of polytopes with a rich combinatorial structure and strong connections to optimization as discussed by the authors, and their antipode is the alternating sum of the faces of a polytope.

Abstract: Generalized permutahedra are a family of polytopes with a rich combinatorial structure and strong connections to optimization. We prove that they are the universal family of polyhedra with a certain Hopf algebraic structure. Their antipode is remarkably simple: the antipode of a polytope is the alternating sum of its faces. Our construction provides a unifying framework to organize numerous combinatorial structures, including graphs, matroids, posets, set partitions, linear graphs, hypergraphs, simplicial complexes, building sets, and simple graphs. We highlight three applications: 1. We obtain uniform proofs of numerous old and new results about the Hopf algebraic and combinatorial structures of these families. In particular, we give the optimal formula for the antipode of graphs, posets, matroids, hypergraphs, and building sets, and we answer questions of Humpert--Martin and Rota. 2. We show that the reciprocity theorems of Stanley and Billera--Jia--Reiner on chromatic polynomials of graphs, order polynomials of posets, and BJR-polynomials of matroids are instances of the same reciprocity theorem for generalized permutahedra. 3. We explain why the formulas for the multiplicative and compositional inverses of power series are governed by the face structure of permutahedra and associahedra, respectively, answering a question of Loday. Along the way, we offer a combinatorial user's guide to Hopf monoids.