Proceedings of the 28-th International Conference on Formal Power Series and Algebraic Combinatorics
4-8 Jul 2016 Vancouver, British Columbia (Canada)
The facial weak order in finite Coxeter groups
Aram Dermenjian  1, 2  , Christophe Hohlweg  2  , Vincent Pilaud  1  
1 : Laboratoire dínformatique de l\'École polytechnique [Palaiseau]  (LIX)  -  Website
Ecole Polytechnique, Centre National de la Recherche Scientifique : UMR7161
Route de Saclay 91128 PALAISEAU CEDEX -  France
2 : Laboratoire de combinatoire et d'Informatique mathématique [Montréal]  (LaCIM)  -  Website
LaCIM Pavillon Président-Kennedy 201, Président-Kennedy, 4ème étage Montréal (Québec) H2X 3Y7 - Adresse postale : CP 8888, Succ. Centre-ville Montréal (Québec) H3C 3P8 -  Canada

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. 

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