Trimness of Closed Intervals in Cambrian Semilattices

Abstract: In this article, we give a short algebraic proof that all closed intervals in a $\gamma$-Cambrian semilattice $\mathcal{C}{\gamma}$ are trim for any Coxeter group $W$ and any Coxeter element $\gamma\in W$. 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 $\mathcal{C}{\gamma}$ is distributive. This problem was open for any Coxeter group that is not a Weyl group.