Séminaire Lotharingien de Combinatoire, 82B.3 (2019), 12 pp.

Vincent Pilaud

Polytopal realizations and Hopf algebra structures for lattice quotients of the weak order

Abstract. Noncrossing arc diagrams provide a powerful combinatorial model for the equivalence classes of the lattice congruences of the weak order on permutations. In this extended abstract, we use these models to construct geometric and algebraic structures on weak order quotients. On the geometric side, we construct, for any given congruence, a polytope whose normal fan is the quotient fan obtained by gluing together the cones of the braid fan that belong to the same congruence class. On the algebraic side, we define Hopf algebra structures which extend classical structures, including the Malvenuto-Reutenauer algebra, the Loday-Ronco algebra, and the Cambrian algebra.


Received: November 15, 2018. Accepted: February 17, 2019. Final version: April 1, 2019.

The following versions are available: