Trimness of Closed Intervals in Cambrian Semilattices
Abstract
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.
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