Brick polytopes, lattices and Hopf algebras

Abstract : Generalizing the connection between the classes of the sylvester congruence and the binary trees, we show that the classes of the congruence of the weak order 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], are in bijection with acyclic k-triangulations of the (n + 2k)-gon, or equivalently with acyclic pipe dreams for the permutation (1,...,k,n + k,...,k + 1,n + k + 1,...,n + 2k). It enables us to transport the known lattice and Hopf algebra structures from the congruence classes of ≡k to these acyclic pipe dreams, and to describe the product and coproduct of this algebra in terms of pipe dreams. Moreover, it shows that the fan obtained by coarsening the Coxeter fan according to the classes of ≡k is the normal fan of the corresponding brick polytope
Keywords : Combinatorics
Document type :
Conference papers
Complete list of metadatas

Cited literature [19 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-02166338
Contributor : Ccsd Sciencesconf.Org <>
Submitted on : Wednesday, June 26, 2019 - 4:59:56 PM
Last modification on : Tuesday, November 5, 2019 - 3:24:02 PM

Identifiers

  • HAL Id : hal-02166338, version 1

Collections

Citation

Vincent Pilaud. Brick polytopes, lattices and Hopf algebras. 28-th International Conference on Formal Power Series and Algebraic Combinatorics, Simon Fraser University, Jul 2016, Vancouver, Canada. ⟨hal-02166338⟩

Share

Metrics

Record views

30

Files downloads

18