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Transactions of the American Mathematical Society

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The facial weak order and its lattice quotients


Authors: Aram Dermenjian, Christophe Hohlweg and Vincent Pilaud
Journal: Trans. Amer. Math. Soc. 370 (2018), 1469-1507
MSC (2010): Primary 05E99, 20F55; Secondary 06B99, 03G10
DOI: https://doi.org/10.1090/tran/7307
Published electronically: October 24, 2017
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Abstract: We investigate the facial weak order, 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 first provide three characterizations of this poset: the original one in terms of cover relations, the geometric one that generalizes the notion of inversion sets, and the combinatorial one as an induced subposet of the poset of intervals of the weak order. These characterizations are then used to show that the facial weak order is in fact a lattice, generalizing a well-known result of A. Björner 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.


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Additional Information

Aram Dermenjian
Affiliation: LIX, École Polytechnique, Palaiseau & LaCIM, Université du Québec À Montréal (UQAM), Montréal, Québec H2H 2A9, Canada
Email: aram.dermenjian@gmail.com

Christophe Hohlweg
Affiliation: LaCIM, Université du Québec À Montréal (UQAM), Montréal, Québec H2H 2A9, Canada
Email: hohlweg.christophe@uqam.ca

Vincent Pilaud
Affiliation: CNRS & LIX, École Polytechnique, Palaiseau, France
Email: vincent.pilaud@lix.polytechnique.fr

DOI: https://doi.org/10.1090/tran/7307
Keywords: Permutahedra, weak order, Coxeter complex, lattice quotients, associahedra, Cambrian lattices.
Received by editor(s): February 16, 2016
Received by editor(s) in revised form: May 18, 2017
Published electronically: October 24, 2017
Additional Notes: The first author was partially supported by the French ANR grant EGOS (12JS0200201) and an ISM Graduate Scholarship. The second author was supported by NSERC Discovery grant Coxeter groups and related structures. The third author was partially supported by the French ANR grants EGOS (12JS0200201) and SC3A (15CE40000401).
Article copyright: © Copyright 2017 American Mathematical Society

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