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, $\mathcal{K}$ and $\overline{\mathcal{K}}$, that map a Hopf monoid $\mathbf{H}$ to graded Hopf algebras $\mathcal{K}\left(\mathbf{H}\right)$ and $\overline{\mathcal{K}}\left(\mathbf{H}\right)$, respectively. There is a natural Hopf monoid structure on linear orders $\mathbf{L}$, and the two Fock functors are related by $\mathcal{K}\left(\mathbf{H}\right)=\overline{\mathcal{K}}(\mathbf{H}\times \mathbf{L})$. Unlike the functor $\overline{\mathcal{K}}$, the functor $\mathcal{K}$ applied to $\mathbf{H}$ may not preserve the antipode of $\mathbf{H}$. In view of the relation between $\mathcal{K}$ and $\overline{\mathcal{K}}$, 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 $H=\mathcal{K}\left(\mathbf{H}\right)$. 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 $\overline{\mathcal{K}}\left(\mathbf{H}\right)$ 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.

Revised:

Accepted:

Published online:

DOI: https://doi.org/10.5802/alco.53

Classification: 16T30, 05E15, 16T05, 18D35

Keywords: Antipode, Hopf monoid, Hopf algebra, combinatorial identities, colorings, hypergraphs, orientations

@article{ALCO_2019__2_5_903_0, author = {Benedetti, Carolina and Bergeron, Nantel}, title = {The antipode of linearized {Hopf} monoids}, journal = {Algebraic Combinatorics}, pages = {903--935}, publisher = {MathOA foundation}, volume = {2}, number = {5}, year = {2019}, doi = {10.5802/alco.53}, mrnumber = {4023571}, language = {en}, url = {https://alco.centre-mersenne.org/articles/10.5802/alco.53/} }

Benedetti, Carolina; Bergeron, Nantel. The antipode of linearized Hopf monoids. Algebraic Combinatorics, Volume 2 (2019) no. 5, pp. 903-935. doi : 10.5802/alco.53. https://alco.centre-mersenne.org/articles/10.5802/alco.53/

[1] Hopf monoid of generalized permutahedra (2017) (https://arxiv.org/abs/1709.07504)

[2] Combinatorial Hopf algebras and generalized Dehn–Sommerville relations, Compositio Mathematica, Volume 142 (2006), pp. 1-30 | Article | MR 2196760 | Zbl 1092.05070

[3] Hopf monoids from class functions on unitriangular matrices, Algebra and Number Theory, Volume 7 (2013) no. 7, pp. 1743-1779 | Article | MR 3117506 | Zbl 1276.05127

[4] Monoidal functors, species and Hopf algebras, CRM Monograph Series, 29, American Mathematical Society, Providence, RI, 2010, 784 pages | MR 2724388 | Zbl 1209.18002

[5] Counting $(\mathbf{3}+\mathbf{1})$-avoiding permutations, European Journal of Combinatorics, Volume 33 (2012) no. 1, pp. 49-61 | Article | MR 2854630 | Zbl 1230.05015

[6] New Invariants for Permutations, Orders and Graphs, 2018 (draft)

[7] The antipode and primitive elements in the Hopf monoid of supercharacters, Journal of Algebraic Combinatorics, Volume 40 (2014) no. 4, pp. 903-938 | Article | MR 3273396 | Zbl 1304.05147

[8] Hypergraphic polytopes: combinatorial properties and antipode (2017) (https://arxiv.org/abs/1712.08848) | Zbl 1411.05192

[9] Combinatorial Hopf Algebras of Simplicial Complexes, SIAM Journal on Discrete Mathematics, Volume 30 (2016) no. 3, pp. 1737-1757 | Article | MR 3543152 | Zbl 1350.16026

[10] Antipodes and involutions, J. Combin. Theory Ser. A, Volume 148 (2017), pp. 275-315 | Article | MR 3603322 | Zbl 1376.16035

[11] Combinatorial species and tree-like structures, Encyclopedia of Mathematics and its Applications, 67, Cambridge University Press, Cambridge, 1998, xx+457 pages (Translated from the 1994 French original by Margaret Readdy, With a foreword by Gian-Carlo Rota) | MR 1629341

[12] A Hopf algebra of subword complexes, Advances in Mathematics, Volume 305 (2017), pp. 1163-1201 | Article | MR 3570156 | Zbl 1359.16029

[13] The Hopf algebras of symmetric functions and quasi-symmetric functions in non-commutative variables are free and co-free, J. Algebra Appl., Volume 8 (2009) no. 4, pp. 581-600 | Article | MR 2555523 | Zbl 1188.16030

[14] Combinatorial Hopf algebras in quantum field theory. I, Rev. Math. Phys., Volume 17 (2005) no. 8, pp. 881-976 | Article | MR 2167639 | Zbl 1090.16016

[15] Hopf Algebras in Combinatorics (2014) (preprint of notes: https://arxiv.org/abs/1409.8356)

[16] The Incidence Hopf Algebra of Graphs, SIAM Journal on Discrete Mathematics, Volume 26 (2012) no. 2, pp. 555-570 | Article | MR 2967484 | Zbl 1256.16020

[17] Hypergraphs and regularity of square-free monomial ideals, Internat. J. Algebra Comput., Volume 23 (2013) no. 7, pp. 1573-1590 | MR 3143595 | Zbl 1285.13017

[18] Strong forms of linearization for Hopf monoids in species, Journal of Algebraic Combinatorics, Volume 42 (2015) no. 2, pp. 391-428 | Article | MR 3369562 | Zbl 1319.05149

[19] On descent algebras and twisted bialgebras, Mosc. Math. J., Volume 4 (2004) no. 1, pp. 199-216 | Article | MR 2074989 | Zbl 1103.16026

[20] A Symmetric Function Generalization of the Chromatic Polynomial of a Graph, Advances in Mathematics, Volume 111 (1995) no. 1, pp. 166-194 | Article | MR 1317387 | Zbl 0831.05027

[21] Increasing and decreasing subsequences and their variants, Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006-2007 European Mathematical Society (2007) | MR 2334203 | Zbl 1133.05002