Towards m-Cambrian Lattices (1308.4813v3)

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 $n$.