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Article Dans Une Revue Journal of Combinatorial Theory, Series A Année : 2018

Brick polytopes, lattice quotients, and Hopf algebras

Résumé

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 $\mathfrak{S}_n$ defined as the transitive closure of the rewriting rule$U ac V_1 b_1 \cdots V_k b_k W \equiv^k U ca V_1 b_1 \cdots V_k b_k W$ 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. 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.
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Dates et versions

hal-02343363 , version 1 (02-11-2019)

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Vincent Pilaud. Brick polytopes, lattice quotients, and Hopf algebras. Journal of Combinatorial Theory, Series A, 2018, 155, pp.418-457. ⟨10.1016/j.jcta.2017.11.014⟩. ⟨hal-02343363⟩
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