Plane posets, special posets, and permutations
Lo\"ic Foissy (LMR)

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 of special posets introduced by Malvenuto and Reutenauer and the Hopf algebra morphism from to to the Hopf algebra of free quasi-symmetric functions given by linear extensions. In particular, we construct two Hopf subalgebras both isomorphic to ; 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 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.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Algebraic structures and combinatorial models · Advanced Topics in Algebra
