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.

60 pages, 43 figures. Version 2: New Part 3 on Schröder Cambrian Algebra. The title change reflects this modification

References in corpus (6)

• Matroid polytopes, nested sets and Bergman fans
• Algebraic and combinatorial structures on pairs of twin binary trees
• Cambrian Hopf Algebras
• Hopf Algebras of m-permutations, (m+1)-ary trees, and m-parking functions
• m-dendriform algebras
• A lattice of combinatorial Hopf algebras, Application to binary trees with multiplicities

Cited by in corpus (26)

• Cambrian Hopf Algebras
• Brick polytopes, lattice quotients, and Hopf algebras
• Quotientopes
• The facial weak order and its lattice quotients
• Compatibility fans for graphical nested complexes
• Associahedra for finite type cluster algebras and minimal relations between $\mathbf{g}$-vectors
• The weak order on integer posets
• Nonsymmetric operads in combinatorics
• Removahedral congruences versus permutree congruences
• Celebrating Loday's Associahedron
• Hopf dreams and diagonal harmonics
• Hopf algebras on decorated noncrossing arc diagrams
• Shard polytopes
• A Note on Flips in Diagonal Rectangulations
• Acyclic reorientation lattices and their lattice quotients
• Linear compactness and combinatorial bialgebras
• The Hopf algebra of integer binary relations
• Markov Chains from Descent Operators on Combinatorial Hopf Algebras
• Hochschild polytopes
• Cambrian triangulations and their tropical realizations
• Geometric realizations of the $s$-weak order and its lattice quotients
• Operads in algebraic combinatorics
• Coxeter-biCatalan combinatorics
• Permutree sorting
• Separating trees and simple congruences of the weak order
• Characterizations of the Borel triangle and Borel polynomials