Brick polytopes, lattice quotients, and Hopf algebras

This paper is motivated by the interplay between the Tamari lattice, J.-L. Loday’s realization of the associahedron, and J.-L. Loday and M. Ronco’s Hopf algebra on binary trees. We show that these constructions extend in the world of acyclic k-triangulations, which were already considered as the vertices of V. Pilaud and F. Santos’ brick polytopes. We describe combinatorially a natural surjection from the permutations to the acyclic k-triangulations. We show that the fibers of this surjection are the classes of the congruence ≡k on Sn defined as the transitive closure of the rewriting rule UacV1b1 · · ·VkbkW ≡k UcaV1b1 · · ·VkbkW for letters a < b1, . . . , bk < c and words U, V1, . . . , Vk,W on [n]. We then show that the increasing flip order on k-triangulations is the lattice quotient of the weak order by this congruence. Moreover, we use this surjection to define a Hopf subalgebra of C. Malvenuto and C. Reutenauer’s Hopf algebra on permutations, indexed by acyclic k-triangulations, and to describe the product and coproduct in this algebra and its dual in term of combinatorial operations on acyclic k-triangulations. Finally, we extend our results in three directions, describing a Cambrian, a tuple, and a Schröder version of these constructions.