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# On the Topology of the Cambrian Semilattices

Abstract : For an arbitrary Coxeter group $W$, David Speyer and Nathan Reading defined Cambrian semilattices $C_{\gamma}$ as certain sub-semilattices of the weak order on $W$. In this article, we define an edge-labelling using the realization of Cambrian semilattices in terms of $\gamma$-sortable elements, and show that this is an EL-labelling for every closed interval of $C_{\gamma}$. In addition, we use our labelling 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 Nathan Reading.
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https://hal.inria.fr/hal-01229735
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Submitted on : Tuesday, November 17, 2015 - 10:20:48 AM
Last modification on : Thursday, July 4, 2019 - 3:30:08 PM
Long-term archiving on: : Friday, April 28, 2017 - 10:08:56 PM

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dmAS0144.pdf
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### Citation

Myrto Kallipoliti, Henri Mühle. On the Topology of the Cambrian Semilattices. 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), 2013, Paris, France. pp.515-526, ⟨10.46298/dmtcs.2320⟩. ⟨hal-01229735⟩

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