Brick polytopes, lattice quotients, and Hopf algebras

Vincent Pilaud

doi:10.1016/j.jcta.2017.11.014

paper

Brick polytopes, lattice quotients, and Hopf algebras

arXiv:1505.07665 · doi:10.1016/j.jcta.2017.11.014

Abstract

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 $\equiv^{k}$ on $S_{n}$ defined as the transitive closure of the rewriting rule $U a c V_{1} b_{1} \dots V_{k} b_{k} W \equiv^{k} U c a V_{1} b_{1} \dots V_{k} b_{k} W$ for letters $a < b_{1}, \dots, b_{k} < c$ and words $U, V_{1}, \dots, V_{k}, W$ on $[n]$ . We then show that the increasing flip order on $k$ -triangulations is the lattice quotient of the weak order by this congruence. 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.

59 pages, 32 figures

Brick polytopes, lattice quotients, and Hopf algebras

Abstract

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