Combinatorial Hopf Algebras of Simplicial Complexes

Abstract

International audience We consider a Hopf algebra of simplicial complexes and provide a cancellation-free formula for its antipode. We then obtain a family of combinatorial Hopf algebras by defining a family of characters on this Hopf algebra. The characters of these Hopf algebras give rise to symmetric functions that encode information about colorings of simplicial complexes and their $f$-vectors. We also use characters to give a generalization of Stanley’s $(-1)$-color theorem. Nous considérons une algèbre de Hopf de complexes simpliciaux et fournissons une formule sans multiplicité pour son antipode. On obtient ensuite une famille d'algèbres de Hopf combinatoires en définissant une famille de caractères sur cette algèbre de Hopf. Les caractères de ces algèbres de Hopf donnent lieu à des fonctions symétriques qui encode de l’information sur les coloriages du complexe simplicial ainsi que son vecteur-$f$. Nousallons également utiliser des caractères pour donner une généralisation du théorème $(-1)$ de Stanley.

Counting faces of graphical zonotopes

“…By applying Theorem 1.1 we obtain the following interpretation of identities elaborated in [2,Propositions 17,19].…”

“…The following modification of the character ζ is considered in [2] in a wider context of the combinatorial Hopf algebra of simplicial complexes. Define ζ q (Γ) = q rk(Γ) , which determines the algebra morphism ζ q : G → k(q), where k(q) is the field of rational functions in q.…”

“…There is a q-analog of the chromatic symmetric function Ψ q (Γ) introduced in a wider context of the combinatorial Hopf algebra of simplicial complexes considered in [2]. It is a symmetric function over the field of rational functions in q.…”

Integer points enumerator of hypergraphic polytopes

“…Example 4.7. Simplicial complexes generate another Hopf subalgebra of HG which is isomorphic to the Hopf algebra of simplicial complexes introduced in [10] and studied more extensively in [3]. It is shown in [1,Lemma 21.2] that hypergraphic polytopes P K and P K 1 corresponding to a simplicial complex K and its 1-skeleton K 1 are normally equivalent and therefore have the same enumerators F q (P K ) = F q (P K 1 ).…”

“…The theory of combinatorial Hopf algebras developed by Aguiar, Bergeron and Sottile in the seminal paper [2] provides an algebraic framework for symmetric and quasisymmetric generating functions arising in enumerative combinatorics. Extensive studies of various combinatorial Hopf algebras are initiated recently [3], [4], [9], [10]. The geometric interpretation of the corresponding (quasi)symmetric functions was first given for matroids [4] and then for simple graphs [6] and building sets [7].…”