The facial weak order and its lattice quotients

Aram Dermenjian, Christophe Hohlweg , Vincent Pilaud

Research output: Contribution to journalArticlepeer-review


We investigate the facial weak order, a poset structure that extends the weak order on a finite Coxeter group to the set of all faces of the permutahedron of . 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.
Original languageEnglish
Pages (from-to)1469-1507
Number of pages39
JournalTransactions of the American Mathematical Society
Issue number10
Publication statusPublished - 24 Oct 2017


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