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.Comment: Final version. The contents of this paper were formerly part of my
now withdrawn submission arXiv:1312.4449. 12 pages, 3 figure