Double posets and the antipode of QSym

Darij Grinberg

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7 Scopus citations


A quasisymmetric function is assigned to every double poset (that is, every finite set endowed with two partial orders) and any weight function on its ground set. This generalizes well-known objects such as monomial and fundamental quasisymmetric functions, (skew) Schur functions, dual immaculate functions, and quasisymmetric (P, ω)-partition enumerators. We prove a formula for the antipode of this function that holds under certain conditions (which are satisfied when the second order of the double poset is total, but also in some other cases); this restates (in a way that to us seems more natural) a result by Malvenuto and Reutenauer, but our proof is new and self-contained. We generalize it further to an even more comprehensive setting, where a group acts on the double poset by automorphisms.

Original languageEnglish (US)
Article number#P2.22
JournalElectronic Journal of Combinatorics
Issue number2
StatePublished - May 5 2017

Bibliographical note

Publisher Copyright:
© 2017, Australian National University. All rights reserved.


  • Antipodes
  • Double posets
  • Hopf algebras
  • P-partitions
  • Posets
  • Quasisymmetric functions


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