Structural Properties of the Cambrian Semilattices -- Consequences of Semidistributivity

Abstract

The $\gamma$-Cambrian semilattices ${\mathcal{C}}_{\gamma }$ defined by Reading and Speyer are a family of meet-semilattices associated with a Coxeter group $W$ and a Coxeter element $\gamma \in W$, and they are lattices if and only if $W$ is finite. In the case where $W$ is the symmetric group ${\mathfrak{S}}_n$ and $\gamma$ is the long cycle $(12\dots n)$ the corresponding $\gamma$-Cambrian lattice is isomorphic to the well-known Tamari lattice ${\mathcal{T}}_n$. Recently, Kallipoliti and the author have investigated ${\mathcal{C}}_{\gamma }$ from a topological viewpoint, and showed that many properties of the Tamari lattices can be generalized nicely. In the present article this investigation is continued on a structural level using the observation of Reading and Speyer that ${\mathcal{C}}_{\gamma }$ is semidistributive. First we prove that every closed interval of ${\mathcal{C}}_{\gamma }$ is a bounded-homomorphic image of a free lattice (in fact it is a so-called $\mathcal{H}\mathcal{H}$-lattice). Subsequently we prove that each closed interval of ${\mathcal{C}}_{\gamma }$ is trim, we determine its breadth, and we characterize the closed intervals that are dismantlable.

This paper has been withdrawn by the author due to a gap in the proof of Theorem 1.1(i). The results in Theorems 1.1(ii)-(iv) and 1.2, and those needed for their proofs remain true, and will be addressed in separate articles. I suspect that the claim of Theorem 1.1(i) is still true. In fact, I suspect that quotients of HH-lattices are HH-lattices again. Comments are very welcome

References in corpus (1)

• Sortable elements and Cambrian lattices