Cancelation Free Formula for the Antipode of Linearized Hopf Monoid

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). Unlike the functor K, the functor K applied to H may not preserve the antipode of H. In this case, one needs to consider the larger Hopf monoid L×H to getH = K(H) = K(L×H) and study the antipode in L × H. One of the main results in this paper provides a cancelation free and multiplicity free formula for the antipode of L × H. From this formula we obtain a new antipode formula for H . We also explore the case when H is commutative and cocommutative. In this situation we get new antipode formulas that despite of not being cancelation free, can be used to obtain one for K(H) in some cases. We recover as well many of the well-known cancelation free formulas in the literature. One of our formulas for computing the antipode in H involves acyclic orientations of hypergraphs as the central tool. In this vein, we obtain polynomials analogous to the chromatic polynomial of a graph, and also identities parallel to Stanley’s (-1)-color theorem. One of our examples introduces a chromatic polynomial for permutations which counts increasing sequences of the permutation satisfying a pattern. We also study the statistic obtained after evaluating such polynomial at −1. Finally, we sketch q deformations and geometric interpretations of our results. This last part will appear in a sequel paper in joint work with J. Machacek. Introduction Computing antipode formulas in various graded Hopf algebras is a classical yet difficult problem to solve. Recently, numerous results in this direction have been provided for various families of Hopf algebras [13, 8, 9, 1, 11, 6]. A motivation to find such formulas lies in their potential geometric interpretation (see for example [1]), or in their use to derive information regarding combinatorial invariants of the discrete objects in play. One example of this is the Hopf algebra of graphs G (see, for instance [13]). In [13] the authors derive the antipode formula and use it to obtain the celebrated Stanley’s (−1)-color theorem: the chromatic polynomial of a graph evaluated at −1 is, up to a sign, the number of acyclic orientations of the graph. On the geometric side, a remarkable result in[1] shows that such antipode is encoded in the f -vector of the graphical zonotope corresponding to the given graph. The general principle is that antipode formulas provide interesting identities for the combinatorial invariants. One of the key results in the theory of Combinatorial Hopf algebras (CHAs) gives us a canonical way of constructing combinatorial invariants with Date: November 8, 2016. 2010 Mathematics Subject Classification. 16T30; 05E15; 16T05; 18D35.