Combinatorial Hopf Algebras of Simplicial Complexes

International audienceWe 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

A Hopf algebra of subword complexes (Extended abstract)

International audienceWe introduce a Hopf algebra structure of subword complexes, including both finite and infinite types. We present an explicit cancellation free formula for the antipode using acyclic orientations of certain graphs, and show that this Hopf algebra induces a natural non-trivial sub-Hopf algebra on c-clusters in the theory of cluster algebras

On the cancellation-free antipode formula for the Malvenuto-Reutenauer Hopf Algebra

For the Malvenuto-Reutenauer Hopf algebra of permutations, we provide a cancellation-free antipode formula for any permutation of the form $ab1\cdots(b-1)(b+1)\cdots(a-1)(a+1)\cdots n$, which starts with the decreasing sequence $ab$ and ends with the increasing sequence $1\cdots(b-1)(b+1)\cdots(a-1)(a+1)\cdots n$, where $1\leq b<a\leq n$. As a consequence, we confirm two conjectures posed by Carolina Benedetti and Bruce E. Sagan.Comment: 19 page