Trimness of closed intervals in Cambrian semilattices
Henri Mühle · Comptes Rendus Mathématique · 2016
In this article, we give a short algebraic proof that all closed intervals in a γ -Cambrian semilattice C γ 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 C γ is distributive. This problem was open for any Coxeter group that is not a Weyl group.