Trimness of closed intervals in Cambrian semilattices

Mühle, Henri

Comptes Rendus. Mathématique, Tome 354 (2016) no. 2, pp. 113-120.

Dans cet article, nous donnons une démonstration courte et algébrique du fait que tous les intervalles bornés d'un demi-treillis γ-cambrien Cγ sont sveltes pour tout groupe de Coxeter W et tout élément de Coxeter γ∈W. Cela signifie que, si un tel intervalle a pour longueur k, il existe une chaîne de longueur k consistant en éléments modulaires à gauche, et il y a exactement k éléments sup-irréductibles et k éléments inf-irréductibles. En conséquence, il s'ensuit que chaque intervalle gradué est distributif. Ce problème était ouvert pour tout groupe de Coxeter qui n'est pas un groupe de Weyl.

In this article, we give a short algebraic proof that all closed intervals in a γ-Cambrian semilattice Cγ are trim for any Coxeter group W and any Coxeter element γ∈W. This means that if such an interval has length k, then there exists a maximal chain of length k consisting of left-modular elements, and there are precisely k join- and k meet-irreducible elements in this interval. Consequently, every graded interval in Cγ is distributive. This problem was open for any Coxeter group that is not a Weyl group.

Mots clés : Cambrian semilattice, Tamari lattice, Coxeter group, Sortable elements, Trimness

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