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. An error in Section 7 has been noticed by Darij Grinberg, and the text has been modified accordingly.

Mots clés: 17A30; 16W30; 05A05; Special posets; permutations; self-dual Hopf algebras; duplicial algebras; den- driform algebras AMS classification 06A11

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

Domaines: Anneaux et algèbres [math.RA]

Loïc Foissy. Plane posets, special posets, and permutations. Advances in Mathematics, 2013. ⟨hal-00619299v4⟩