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The absolute order is a natural partial order on a Coxeter group W. It can be viewed as an analogue of the weak order on W in which the role of the generating set of simple reflections in W is played by the set of all reflections in W. By use of a notion of constructibility for partially ordered sets, it is proved that the absolute order on the symmetric(More)
In this paper we present a bijection between two well known families of Catalan objects: the set of facets of the m-generalized cluster complex ∆ m (A n) and that of dominant regions in the m-Catalan arrangement Cat m (A n), where m ∈ N >0. In particular, the map which we define bijects facets containing the negative simple root −α to dominant regions(More)
For an arbitrary Coxeter group W , David Speyer and Nathan Reading defined Cambrian semilattices Cγ as certain sub-semilattices of the weak order on W. In this article, we define an edge-labeling using the realization of Cambrian semilattices in terms of γ-sortable elements, and show that this is an EL-labeling for every closed interval of Cγ. In addition,(More)
In this paper we study topological properties of the poset of injective words and the lattice of classical non-crossing partitions. Specifically, it is shown that after the removal of the bottom and top elements (if existent) these posets are doubly Cohen-Macaulay. This extends the well-known result that those posets are shellable. Both results rely on a(More)
The absolute order is a natural partial order on a Coxeter group W. It can be viewed as an analogue of the weak order on W in which the role of the generating set of simple reflections in W is played by the set of all reflections in W. By use of a notion of constructibility for partially ordered sets, it is proved that the absolute order on the symmetric(More)
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