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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