Trimness of Closed Intervals in Cambrian Semilattices

TL;DR

This paper proves that all closed intervals in $ ext{γ}$-Cambrian semilattices are trim, ensuring they have well-structured chains and distributive properties, which was previously unknown for non-Weyl Coxeter groups.

Contribution

Provides a short algebraic proof that all closed intervals in $ ext{γ}$-Cambrian semilattices are trim for any Coxeter group and element, extending known results beyond Weyl groups.

Findings

All closed intervals in $ ext{γ}$-Cambrian semilattices are trim.

Every graded interval in these semilattices is distributive.

The result holds for any Coxeter group and Coxeter element.

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.