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 S_n defined as the transitive closure of the rewriting rule UacV₁b₁⋯V_kb_kW≡^kUcaV₁b₁⋯V_kb_kW for letters a<b₁,…,b_k<c and words U,V₁,…,V_k,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. Finally, 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 in term of combinatorial operations on acyclic k-triangulations.