The facial weak order and its lattice quotients

TL;DR

This paper introduces the facial weak order, a lattice structure extending the weak order on Coxeter groups to all faces of the permutahedron, with geometric and combinatorial characterizations and applications to cube and associahedron faces.

Contribution

It provides a new lattice structure called the facial weak order, generalizes known results, and links lattice congruences to geometric face classifications.

Findings

Facial weak order is a lattice extending the weak order.

Characterizations of the facial weak order are geometric and combinatorial.

Lattice congruences of the weak order induce congruences of the facial weak order.

Abstract

We investigate a poset structure that extends the weak order on a finite Coxeter group $W$ to the set of all faces of the permutahedron of $W$. We call this order the facial weak order. We first provide two alternative characterizations of this poset: a first one, geometric, that generalizes the notion of inversion sets of roots, and a second one, combinatorial, that uses comparisons of the minimal and maximal length representatives of the cosets. These characterizations are then used to show that the facial weak order is in fact a lattice, generalizing a well-known result of A. Bj\"orner for the classical weak order. Finally, we show that any lattice congruence of the classical weak order induces a lattice congruence of the facial weak order, and we give a geometric interpretation of their classes. As application, we describe the facial boolean lattice on the faces of the cube and the facial Cambrian lattice on the faces of the corresponding generalized associahedron.