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.

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 The contents of this paper were formerly part of my now withdrawn submission arXiv:1312.4449. 12 pages, 3 figures

References in corpus (3)

• Noncrossing partitions and representations of quivers
• Sortable elements and Cambrian lattices
• Sortable elements in infinite Coxeter groups

Cited by in corpus (2)

• Rowmotion in slow motion
• SB-Labelings, Distributivity, and Bruhat Order on Sortable Elements