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. Bjo ̈rner 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 its classes.
Highlights
The Cayley graph of a Coxeter system (W, S) is naturally oriented by the weak order on W : an edge is oriented from w to ws if s ∈ S is such that (w)
The weak order is a very useful tool to study Coxeter groups as it encodes the combinatorics of reduced words associated to (W, S), and it underlines the connection between the words and the root system via the notion of inversion sets, see for instance [Dye11, HL16] and the references therein
We study a poset structure on all faces of the W -permutahedron that we call the facial weak order
Summary
We start by fixing notations and classical definitions on finite Coxeter groups. Details can be found in textbooks by J. The reader familiar with finite Coxeter groups and root systems is invited to proceed directly to Section 2
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