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
Highlights
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 U acV1b1 · · · VkbkW ≡k U caV1b1 · · · 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)
Reading in his work on Lattice congruences, fans and Hopf algebras [Rea05]. He proves that any lattice congruence ≡ of the weak order on Sn defines a complete simplicial fan F≡ refined by the Coxeter fan, and he characterizes in terms of simple
His work opens two natural questions. It is not clear which of the fans F≡ are normal fans of polytopes, as in the previous example of the associahedron. This construction produces a combinatorial Hopf algebra whose basis is indexed by the congruence classes of (≡n)n∈N
Summary
We recall the notion of flips in pipe dreams and study the graph of increasing flips in acyclic k-twists. Proposition 26 The cones C♦(τ ) | τ ∈ Sn , C♦(T) | T ∈ AT k(n) , and C♦(O) | O ∈ AOk(n) , together with all their faces, respectively form the normal fans of the permutahedron Perm(n), of the brick polytope Brickk(n) and of the zonotope Zonok(n). Using these normal fans, one can interpret geometrically the maps insk, cank, and reck as follows.
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