Geodesic
The Antipode of Linearized Hopf Monoids
Séminaire lotharingien de combinatoire, 78B (2017)

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Many combinatorial Hopf algebras H in the literature are the functorial image of a linearized Hopf monoid H. That is H = K(H) or H = K-(H). For the functor K the antipode of H may not be preserved, but the Hopf monoid L x H gives H = K(H) = K-(L x H) and the functor K- preserves antipodes. In this paper, we give a cancelation free and multiplicity free formula for the antipode of L x H. We also compute the antipode for H when it is commutative and cocommutative. We get new formulas that are not always cancelation free but can be used to obtain one for H in some cases. The formulas for H involve acyclic orientations of hypergraphs. In an example, we introduce a chromatic invariant for the increasing sequences of a permutation and show that its evaluation at t = -1 relates to another statistic on permutations. 

@article{SLC_2017_78B_a12,
     author = {Carolina Benedetti and Nantel Bergeron},
     title = {The {Antipode} of {Linearized} {Hopf} {Monoids}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {78B},
     year = {2017},
     url = {http://geodesic.mathdoc.fr/item/SLC_2017_78B_a12/}
}
TY  - JOUR
AU  - Carolina Benedetti
AU  - Nantel Bergeron
TI  - The Antipode of Linearized Hopf Monoids
JO  - Séminaire lotharingien de combinatoire
PY  - 2017
VL  - 78B
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SLC_2017_78B_a12/
ID  - SLC_2017_78B_a12
ER  - 
%0 Journal Article
%A Carolina Benedetti
%A Nantel Bergeron
%T The Antipode of Linearized Hopf Monoids
%J Séminaire lotharingien de combinatoire
%D 2017
%V 78B
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SLC_2017_78B_a12/
%F SLC_2017_78B_a12
Carolina Benedetti; Nantel Bergeron. The Antipode of Linearized Hopf Monoids. Séminaire lotharingien de combinatoire, 78B (2017). http://geodesic.mathdoc.fr/item/SLC_2017_78B_a12/