On the Topology of the Cambrian Semilattices

  • Myrto Kallipoliti
  • Henri Mühle
Keywords: EL-Shellability, Möbius Function, Cambrian Semilattice, Weak Order, Coxeter Group


For an arbitrary Coxeter group $W$, Reading and Speyer defined Cambrian semilattices $\mathcal{C}_{\gamma}$ as sub-semilattices of the weak order on $W$ induced by so-called $\gamma$-sortable elements. In this article, 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.
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