The antipode of linearized Hopf monoids
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 𝒦 ¯, that map a Hopf monoid H to graded Hopf algebras 𝒦(H) and 𝒦 ¯(H), respectively. There is a natural Hopf monoid structure on linear orders L, and the two Fock functors are related by 𝒦(H)=𝒦 ¯(H×L). Unlike the functor 𝒦 ¯, the functor 𝒦 applied to H may not preserve the antipode of H. In view of the relation between 𝒦 and 𝒦 ¯, one may consider instead of H the larger Hopf monoid L×H and study the antipode of L×H. One of the main results in this paper provides a cancellation free and multiplicity free formula for the antipode of L×H. As a consequence, we obtain a new antipode formula for the Hopf algebra H=𝒦(H). We explore the case when 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 𝒦 ¯(H) 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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/alco.53
Classification : 16T30, 05E15, 16T05, 18D35
Mots clés : Antipode, Hopf monoid, Hopf algebra, combinatorial identities, colorings, hypergraphs, orientations
Benedetti, Carolina 1 ; Bergeron, Nantel 2

1 Departamento de Matemáticas Universidad de los Andes Bogotá, Colombia
2 Department of Mathematics and Statistics York University Toronto, Ontario M3J 1P3, Canada
@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 = {http://www.numdam.org/articles/10.5802/alco.53/}
}
TY  - JOUR
AU  - Benedetti, Carolina
AU  - Bergeron, Nantel
TI  - The antipode of linearized Hopf monoids
JO  - Algebraic Combinatorics
PY  - 2019
SP  - 903
EP  - 935
VL  - 2
IS  - 5
PB  - MathOA foundation
UR  - http://www.numdam.org/articles/10.5802/alco.53/
DO  - 10.5802/alco.53
LA  - en
ID  - ALCO_2019__2_5_903_0
ER  - 
%0 Journal Article
%A Benedetti, Carolina
%A Bergeron, Nantel
%T The antipode of linearized Hopf monoids
%J Algebraic Combinatorics
%D 2019
%P 903-935
%V 2
%N 5
%I MathOA foundation
%U http://www.numdam.org/articles/10.5802/alco.53/
%R 10.5802/alco.53
%G en
%F ALCO_2019__2_5_903_0
Benedetti, Carolina; Bergeron, Nantel. The antipode of linearized Hopf monoids. Algebraic Combinatorics, Tome 2 (2019) no. 5, pp. 903-935. doi : 10.5802/alco.53. http://www.numdam.org/articles/10.5802/alco.53/

[1] Aguiar, Marcelo; Ardila, Federico Hopf monoid of generalized permutahedra (2017) (https://arxiv.org/abs/1709.07504)

[2] Aguiar, Marcelo; Bergeron, Nantel; Sottile, Frank Combinatorial Hopf algebras and generalized Dehn–Sommerville relations, Compositio Mathematica, Volume 142 (2006), pp. 1-30 | DOI | MR | Zbl

[3] Aguiar, Marcelo; Bergeron, Nantel; Thiem, Nathaniel Hopf monoids from class functions on unitriangular matrices, Algebra and Number Theory, Volume 7 (2013) no. 7, pp. 1743-1779 | DOI | MR | Zbl

[4] Aguiar, Marcelo; Mahajan, Swapneel Monoidal functors, species and Hopf algebras, CRM Monograph Series, 29, American Mathematical Society, Providence, RI, 2010, 784 pages | MR | Zbl

[5] Atkinson, M. D.; Sagan, Bruce E.; Vatter, Vincent Counting (3+1)-avoiding permutations, European Journal of Combinatorics, Volume 33 (2012) no. 1, pp. 49-61 | DOI | MR | Zbl

[6] Aval, Jean-Christophe; Bergeron, Nantel; Machacek, John New Invariants for Permutations, Orders and Graphs, 2018 (draft)

[7] Baker-Jarvis, Duff; Bergeron, Nantel; Thiem, Nathaniel The antipode and primitive elements in the Hopf monoid of supercharacters, Journal of Algebraic Combinatorics, Volume 40 (2014) no. 4, pp. 903-938 | DOI | MR | Zbl

[8] Benedetti, Carolina; Bergeron, Nantel; Machacek, John Hypergraphic polytopes: combinatorial properties and antipode (2017) (https://arxiv.org/abs/1712.08848) | Zbl

[9] Benedetti, Carolina; Hallam, Joshua; Machacek, John Combinatorial Hopf Algebras of Simplicial Complexes, SIAM Journal on Discrete Mathematics, Volume 30 (2016) no. 3, pp. 1737-1757 | DOI | MR | Zbl

[10] Benedetti, Carolina; Sagan, Bruce E. Antipodes and involutions, J. Combin. Theory Ser. A, Volume 148 (2017), pp. 275-315 | DOI | MR | Zbl

[11] Bergeron, François; Labelle, Gilbert; Leroux, Pierre 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

[12] Bergeron, Nantel; Ceballos, Cesar A Hopf algebra of subword complexes, Advances in Mathematics, Volume 305 (2017), pp. 1163-1201 | DOI | MR | Zbl

[13] Bergeron, Nantel; Zabrocki, Mike 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 | DOI | MR | Zbl

[14] Figueroa, Hector; Gracia-Bondía, Jose M. Combinatorial Hopf algebras in quantum field theory. I, Rev. Math. Phys., Volume 17 (2005) no. 8, pp. 881-976 | DOI | MR | Zbl

[15] Grinberg, Darij; Reiner, Victor Hopf Algebras in Combinatorics (2014) (preprint of notes: https://arxiv.org/abs/1409.8356)

[16] Humpert, Brandon; Martin, Jeremy L. The Incidence Hopf Algebra of Graphs, SIAM Journal on Discrete Mathematics, Volume 26 (2012) no. 2, pp. 555-570 | DOI | MR | Zbl

[17] Lin, Kuei-Nuan; McCullough, Jason Hypergraphs and regularity of square-free monomial ideals, Internat. J. Algebra Comput., Volume 23 (2013) no. 7, pp. 1573-1590 | MR | Zbl

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

[19] Patras, Frédéric; Reutenauer, Christophe On descent algebras and twisted bialgebras, Mosc. Math. J., Volume 4 (2004) no. 1, pp. 199-216 | DOI | MR | Zbl

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

[21] Stanley, Richard P. Increasing and decreasing subsequences and their variants, Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006-2007 European Mathematical Society, Amer. Math. Soc., Providence, RI (2007) | MR | Zbl

Cité par Sources :