Séminaire Lotharingien de Combinatoire, 78B.13 (2017), 12 pp.
Carolina Benedetti and Nantel Bergeron
The Antipode of Linearized Hopf Monoids
Abstract.
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.
Received: November 14, 2016.
Accepted: February 17, 2017.
Final version: April 1, 2017.
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