Cambrian Hopf algebras

Abstract

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 Cambrian 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. Contents ± 57 3.3. Schröder-Cambrian tuples

Quotientopes

“…This congruence has been widely studied in connection to geometry and algebra [7,11,12]. Among many other examples of relevant lattice quotients of the weak order, let us mention the (type A) Cambrian lattices [3,23], the boolean lattice, Figure 1 (colour online). The Hasse diagram of the weak order on S4 (left) can be seen as the oriented dual graph of braid fan (middle) or as an orientation of the graph of the permutahedron Permp4q (right).…”

The Hopf algebra of integer binary relations

“…We refer to [Cha00,CP17] for more details and just provide an example of product and coproduct in this Hopf algebra.…”

Counting with Borel’s triangle

“…Recently another triangular array that is closely related to Catalan's triangle has appeared in various studies in commutative algebra, combinatorics, and discrete geometry. It is the sequence A234950 in OEIS and is called Borel's triangle, which is related to pseudo-triangulations of point sets [1] and the Betti numbers of certain principal Borel ideals [9], and appears in Cambrian Hopf algebras [6], quantum physics [12], and permutation patterns [14]. In the second author's work of parking functions and parking distributions on trees, Borel's triangle gives the coefficients of certain generating functions on the nondecreasing parking functions [5,Section 3], which inspires the project on finding classes of objects that are counted by Borel's triangle and characterizing their combinatorial structures.…”