Plane posets, special posets, and permutations

Details Plane posets, special posets, and permutations Plane posets, special posets, and permutations

2011-09-06

arxiv.org/abs/1109.1101v2

Mathematics

Rings and Algebras

36 pages. Minor modifications, plus reminders on the construction of FQSym

Scientific paper

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.

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