Cambrian Hopf algebras

Grégory Chatel (1) , Vincent Pilaud (2)

1 LIGM - Laboratoire d'Informatique Gaspard-Monge
2 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]

Cambrian trees are oriented and labeled trees which fulfill local conditions around each node generalizing the conditions for classical binary search trees. Based on the bijective correspondence between signed permutations and leveled Cambrian trees, we define the Cam-brian Hopf algebra generalizing J.-L. Loday and M. Ronco's algebra on binary trees. We describe combinatorially the products and coproducts of both the Cambrian algebra and its dual in terms of operations on Cambrian trees. We also define multiplicative bases of the Cambrian algebra and study structural and combinatorial properties of their indecomposable elements. Finally, we extend to the Cambrian setting different algebras connected to binary trees, in particular S. Law and N. Reading's Baxter Hopf algebra on quadrangulations and S. Giraudo's equivalent Hopf algebra on twin binary trees, and F. Chapoton's Hopf algebra on all faces of the associahedron.

Mots clés: trees; combinatorics; lattice; hopf algebra; hopf algebras; Cambrian

Domaines: Combinatoire [math.CO]; Informatique [cs]; Mathématiques [math]

Grégory Chatel, Vincent Pilaud. Cambrian Hopf algebras. 2015. ⟨hal-01283747⟩