The antipode of linearized Hopf monoids

Benedetti, Carolina; Bergeron, Nantel

Algebraic Combinatorics, Tome 2 (2019) no. 5, pp. 903-935.

In this paper, a Hopf monoid is an algebraic structure built on objects in the category of Joyal’s vector species. There are two Fock functors, $𝒦$ and $\overline{𝒦}$, that map a Hopf monoid $\mathbf{H}$ to graded Hopf algebras $𝒦\left(\mathbf{H}\right)$ and $\overline{𝒦}\left(\mathbf{H}\right)$, respectively. There is a natural Hopf monoid structure on linear orders $\mathbf{L}$, and the two Fock functors are related by $𝒦\left(\mathbf{H}\right)=\overline{𝒦}(\mathbf{H}\times \mathbf{L})$. Unlike the functor $\overline{𝒦}$, the functor $𝒦$ applied to $\mathbf{H}$ may not preserve the antipode of $\mathbf{H}$. In view of the relation between $𝒦$ and $\overline{𝒦}$, one may consider instead of $\mathbf{H}$ the larger Hopf monoid $\mathbf{L}\times \mathbf{H}$ and study the antipode of $\mathbf{L}\times \mathbf{H}$. One of the main results in this paper provides a cancellation free and multiplicity free formula for the antipode of $\mathbf{L}\times \mathbf{H}$. As a consequence, we obtain a new antipode formula for the Hopf algebra $H=𝒦\left(\mathbf{H}\right)$. We explore the case when $\mathbf{H}$ is commutative and cocommutative, and obtain new antipode formulas that, although not cancellation free, they can be used to obtain an antipode formula for $\overline{𝒦}\left(\mathbf{H}\right)$ in some cases. We also recover many well-known identities in the literature involving antipodes of certain Hopf algebras. In our study of commutative and cocommutative Hopf monoids, hypergraphs and acyclic orientations play a central role. We obtain polynomials analogous to the chromatic polynomial of a graph, and also identities parallel to Stanley’s ($-1$)-color theorem. An important consequence of our notion of acyclic orientation of hypergraphs is a geometric interpretation for the antipode formula for hypergraphs. This interpretation, which differs from the recent work of Aguiar and Ardila as the Hopf structures involved are different, appears in subsequent work by the authors.

Classification : 16T30, 05E15, 16T05, 18D35
Mots clés : Antipode, Hopf monoid, Hopf algebra, combinatorial identities, colorings, hypergraphs, orientations

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