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The facial weak order and its lattice quotients. (English) Zbl 1375.05270
Summary: 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. Bj√∂rner [Contemp. Math. 34, 175–195 (1984; Zbl 0594.20029)] 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.

05E15 Combinatorial aspects of groups and algebras (MSC2010)
20F55 Reflection and Coxeter groups (group-theoretic aspects)
06A07 Combinatorics of partially ordered sets
05B05 Combinatorial aspects of block designs
03G10 Logical aspects of lattices and related structures
20F65 Geometric group theory
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