Plane posets, special posets, and permutations

Loïc Foissy (1)

1 LMR - Laboratoire de Mathématiques de Reims

We study the self-dual Hopf algebra $\h_{\SP}$ of special posets introduced by Malvenuto and Reutenauer and the Hopf algebra morphism from $\h_{\SP}$ to to the Hopf algebra of free quasi-symmetric functions $\FQSym$ given by linear extensions. In particular, we construct two Hopf subalgebras both isomorphic to $\FQSym$; the first one is based on plane posets, the second one on heap-ordered forests. An explicit isomorphism between these two Hopf subalgebras is also defined with the help of two transformations on special posets. The restriction of the Hopf pairing of $\h_{\SP}$ to these Hopf subalgebras and others is also studied, as well as certain isometries between them. These problems are solved using duplicial and dendriform structures.

Mots clés: Special posets; permutations; self-dual Hopf algebras; duplicial algebras; dendriform algebras.; dendriform algebras

Domaines: Anneaux et algèbres [math.RA]

Loïc Foissy. Plane posets, special posets, and permutations. 2011. ⟨hal-00619299v2⟩