On the Topology of the Cambrian Semilattices

Abstract

For an arbitrary Coxeter group W and a Coxeter element γ ∈ W, Reading and Speyer defined the Cambrian semilattice $\mathcal{C}_{\gamma }$ as the sub-semilattice of the weak order on W induced by so-called γ-sortable elements. In this note, we define an edge-labeling of $\mathcal{C}_{\gamma }$, and show that this is an EL-labeling for every closed interval of $\mathcal{C}_{\gamma }$. In addition, we use our labeling to show that every finite open interval in a Cambrian semilattice is either contractible or spherical, and we characterize the spherical intervals, generalizing a result by Reading.