Browse issues
no. 1
Tridendriform algebras on hypergraph polytopes
Curien, Pierre-Louis1; Delcroix-Oger, Bérénice2; Obradović, Jovana3
1 Université Paris Cité CNRS Inria IRIF Picube project-team F-75006 Paris France
2 Université de Montpellier CNRS IMAG F-34090 Montpellier France
3 Mathematical Institute of the Serbian Academy of Sciences and Arts Belgrade Serbia
Algebraic Combinatorics, Volume 8 (2025) no. 1, pp. 201-234.

We extend the works of Loday–Ronco and Burgunder–Ronco on the tridendriform decomposition of the shuffle product on the faces of associahedra and permutohedra, to other families of hypergraph polytopes (or nestohedra), including simplices, hypercubes and some new families. We also extend the shuffle product to take more than two arguments, and define accordingly a new algebraic structure, that we call polydendriform, from which the original tridendriform equations can be crisply synthesized.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.401
Classification: 18M60, 52B05
Keywords: tridendriform structure, polydendriform structure, associative product, shuffle product, hypergraph polytopes, nestohedra

Curien, Pierre-Louis 1; Delcroix-Oger, Bérénice 2; Obradović, Jovana 3

1 Université Paris Cité CNRS Inria IRIF Picube project-team F-75006 Paris France
2 Université de Montpellier CNRS IMAG F-34090 Montpellier France
3 Mathematical Institute of the Serbian Academy of Sciences and Arts Belgrade Serbia
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{ALCO_2025__8_1_201_0,
     author = {Curien, Pierre-Louis and Delcroix-Oger, B\'er\'enice and Obradovi\'c, Jovana},
     title = {Tridendriform algebras on hypergraph polytopes},
     journal = {Algebraic Combinatorics},
     pages = {201--234},
     publisher = {The Combinatorics Consortium},
     volume = {8},
     number = {1},
     year = {2025},
     doi = {10.5802/alco.401},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.401/}
}
TY  - JOUR
AU  - Curien, Pierre-Louis
AU  - Delcroix-Oger, Bérénice
AU  - Obradović, Jovana
TI  - Tridendriform algebras on hypergraph polytopes
JO  - Algebraic Combinatorics
PY  - 2025
SP  - 201
EP  - 234
VL  - 8
IS  - 1
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.401/
DO  - 10.5802/alco.401
LA  - en
ID  - ALCO_2025__8_1_201_0
ER  - 
%0 Journal Article
%A Curien, Pierre-Louis
%A Delcroix-Oger, Bérénice
%A Obradović, Jovana
%T Tridendriform algebras on hypergraph polytopes
%J Algebraic Combinatorics
%D 2025
%P 201-234
%V 8
%N 1
%I The Combinatorics Consortium
%U https://alco.centre-mersenne.org/articles/10.5802/alco.401/
%R 10.5802/alco.401
%G en
%F ALCO_2025__8_1_201_0
Curien, Pierre-Louis; Delcroix-Oger, Bérénice; Obradović, Jovana. Tridendriform algebras on hypergraph polytopes. Algebraic Combinatorics, Volume 8 (2025) no. 1, pp. 201-234. doi : 10.5802/alco.401. https://alco.centre-mersenne.org/articles/10.5802/alco.401/

[1] Aguiar, Marcelo; Ardila, Federico Hopf monoids and generalized permutahedra, Mem. Amer. Math. Soc., Volume 289 (2023) no. 1437, p. vi+119 | DOI | MR | Zbl

[2] Burgunder, Emily; Ronco, María Tridendriform structure on combinatorial Hopf algebras, J. Algebra, Volume 324 (2010) no. 10, pp. 2860-2883 | DOI | MR | Zbl

[3] Carr, Michael P.; Devadoss, Satyan L. Coxeter complexes and graph-associahedra, Topology Appl., Volume 153 (2006) no. 12, pp. 2155-2168 | DOI | MR | Zbl

[4] Chapoton, Frédéric Algèbres de Hopf des permutahèdres, associahèdres et hypercubes, Adv. Math., Volume 150 (2000) no. 2, pp. 264-275 | DOI | MR | Zbl

[5] Curien, Pierre-Louis; Delcroix-Oger, Bérénice; Obradović, Jovana Restriction, shuffles and order in nestohedra (2024) (manuscript in progress)

[6] Curien, Pierre-Louis; Laplante-Anfossi, Guillaume Term rewriting on nestohedra (2024) | arXiv

[7] Curien, Pierre-Louis; Obradović, Jovana; Ivanović, Jelena Syntactic aspects of hypergraph polytopes, J. Homotopy Relat. Struct., Volume 14 (2019) no. 1, pp. 235-279 | DOI | MR | Zbl

[8] Došen, Kosta; Petrić, Zoran Hypergraph polytopes, Topology Appl., Volume 158 (2011) no. 12, pp. 1405-1444 | DOI | MR | Zbl

[9] Dotsenko, Vladimir; Keilthy, Adam; Lyskov, Denis Reconnectads, Algebr. Comb., Volume 7 (2024) no. 3, pp. 801-842 | MR | Zbl

[10] Feichtner, Eva Maria; Sturmfels, Bernd Matroid polytopes, nested sets and Bergman fans, Port. Math. (N.S.), Volume 62 (2005) no. 4, pp. 437-468 | MR | Zbl

[11] Forcey, Stefan; Ronco, María Algebraic structures on graph associahedra, J. Lond. Math. Soc. (2), Volume 106 (2022) no. 2, pp. 1189-1231 | DOI | MR | Zbl

[12] Giraudo, Samuele Pluriassociative algebras II: The polydendriform operad and related operads, Adv. in Appl. Math., Volume 77 (2016), pp. 43-85 | DOI | MR | Zbl

[13] Loday, Jean-Louis; Ronco, María Trialgebras and families of polytopes, Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic K-theory (Contemp. Math.), Volume 346, Amer. Math. Soc., Providence, RI, 2004, pp. 369-398 | DOI | MR | Zbl

[14] Loday, Jean-Louis; Ronco, María O. Hopf algebra of the planar binary trees, Adv. Math., Volume 139 (1998) no. 2, pp. 293-309 | DOI | MR | Zbl

[15] Loday, Jean-Louis; Ronco, María O. Order structure on the algebra of permutations and of planar binary trees, J. Algebr. Comb., Volume 15 (2002) no. 3, pp. 253-270 | DOI | MR | Zbl

[16] Malvenuto, Claudia; Reutenauer, Christophe Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra, Volume 177 (1995) no. 3, pp. 967-982 | DOI | MR | Zbl

[17] Novelli, Jean-Christophe; Thibon, Jean-Yves Polynomial realizations of some trialgebras, 18th Formal Power Series and Algebraic Combinatorics (FPSAC’06) (2006) no. 1, pp. 243-254 | HAL

[18] Palacios, Patricia; Ronco, María O. Weak Bruhat order on the set of faces of the permutohedron and the associahedron, J. Algebra, Volume 299 (2006) no. 2, pp. 648-678 | DOI | MR | Zbl

[19] Postnikov, Alexander Permutohedra, associahedra, and beyond, Int. Math. Res. Not., Volume 2009 (2009) no. 6, pp. 1026-1106 | DOI | MR | Zbl

[20] Ronco, María Generalized Tamari order, Associahedra, Tamari lattices and related structures (Progr. Math.), Volume 299, Birkhäuser/Springer, Basel, 2012, pp. 339-350 | DOI | MR | Zbl

[21] Zelevinsky, Andrei Nested complexes and their polyhedral realizations, Pure Appl. Math. Q., Volume 2 (2006) no. 3, pp. 655-671 | DOI | MR | Zbl

Cited by Sources: