Geodesic
Double Posets and the Antipode of QSym
Séminaire lotharingien de combinatoire, 78B (2017)

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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. 

@article{SLC_2017_78B_a32,
     author = {Darij Grinberg},
     title = {Double {Posets} and the {Antipode} of {QSym}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {78B},
     year = {2017},
     url = {http://geodesic.mathdoc.fr/item/SLC_2017_78B_a32/}
}
TY  - JOUR
AU  - Darij Grinberg
TI  - Double Posets and the Antipode of QSym
JO  - Séminaire lotharingien de combinatoire
PY  - 2017
VL  - 78B
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SLC_2017_78B_a32/
ID  - SLC_2017_78B_a32
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%0 Journal Article
%A Darij Grinberg
%T Double Posets and the Antipode of QSym
%J Séminaire lotharingien de combinatoire
%D 2017
%V 78B
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SLC_2017_78B_a32/
%F SLC_2017_78B_a32
Darij Grinberg. Double Posets and the Antipode of QSym. Séminaire lotharingien de combinatoire, 78B (2017). http://geodesic.mathdoc.fr/item/SLC_2017_78B_a32/