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Journal Articles Advances in Mathematics Year : 2013

Plane posets, special posets, and permutations

Loïc Foissy


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.
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Dates and versions

hal-00619299 , version 1 (06-09-2011)
hal-00619299 , version 2 (13-04-2012)
hal-00619299 , version 3 (05-04-2013)
hal-00619299 , version 4 (19-06-2020)



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


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