Towards m-Cambrian Lattices

TL;DR

This paper introduces a family of lattices generalizing Cambrian lattices for dihedral groups, explores their structural properties, and conjectures extensions to symmetric groups and $m$-Tamari lattices.

Contribution

It defines new Fuss-Catalan type lattices associated with dihedral groups, proves their key properties, and conjectures their relation to $m$-Tamari lattices for symmetric groups.

Findings

Lattices $ ext{C}_k^{(m)}$ are trim and EL-shellable.

Special case $k=3$ matches $m$-Tamari lattice of parameter 3.

Conjecture relating lattice completion for symmetric groups to $m$-Tamari lattices.

Abstract

For positive integers $m$ and $k$, we introduce a family of lattices $\mathcal{C}_{k}^{(m)}$ associated to the Cambrian lattice $\mathcal{C}_{k}$ of the dihedral group $I_{2}(k)$. We show that $\mathcal{C}_{k}^{(m)}$ satisfies some basic properties of a Fuss-Catalan generalization of $\mathcal{C}_{k}$, namely that $\mathcal{C}_{k}^{(1)}=\mathcal{C}_{k}$ and $\bigl\lvert\mathcal{C}_{k}^{(m)}\bigr\rvert=\mbox{Cat}^{(m)}\bigl(I_{2}(k)\bigr)$. Subsequently, we prove some structural and topological properties of these lattices---namely that they are trim and EL-shellable---which were known for $\mathcal{C}_{k}$ before. Remarkably, our construction coincides in the case $k=3$ with the $m$-Tamari lattice of parameter 3 due to Bergeron and Pr{\'e}ville-Ratelle. Eventually, we investigate this construction in the context of other Coxeter groups, in particular we conjecture that the lattice completion of the analogous construction for the symmetric group $\mathfrak{S}_{n}$ and the long cycle $(1\;2\;\ldots\;n)$ is isomorphic to the $m$-Tamari lattice of parameter $n$.