Plane posets, special posets, and permutations

TL;DR

This paper explores the structure of special posets and their associated Hopf algebras, establishing isomorphisms with free quasi-symmetric functions and analyzing their algebraic properties using advanced combinatorial structures.

Contribution

It constructs two new Hopf subalgebras isomorphic to FQSym based on plane posets and heap-ordered forests, and defines an explicit isomorphism between them.

Findings

Constructed two Hopf subalgebras isomorphic to FQSym

Defined an explicit isomorphism between these subalgebras

Analyzed the Hopf pairing and isometries using duplicial and dendriform structures

Abstract

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.