Abstract
For any finite Coxeter group $W$ of rank $n$ we show that the order complex of the lattice of non-crossing partitions $\mathrm{NC}(W)$ embeds as a chamber subcomplex into a spherical building of type $A_{n-1}$. We use this to give a new proof of the fact that the non-crossing partition lattice in type $A_n$ is supersolvable for all $n$. Moreover, we show that in case $B_n$, this is only the case if $n<4$. We also obtain a lower bound on the radius of the Hurwitz graph $H(W)$ in all types and re-prove that in type $A_n$ the radius is $\binom{n}{2}$. A Corrigendum for this paper was added on May 17, 2018.
Highlights
The lattice of non-crossing partitions NC(W, c) of a finite Coxeter group W, defined with respect to some Coxeter element c, is the interval below c in the absolute order, that is NC(W, c) = {π ∈ W : π ≤ c}
We will prove the mentioned estimates and equalities on the diameter and radius of the Hurwitz graph H(W ). This result relies on the embedding of the Hurwitz graph into the chamber graph Γ∆ of the building ∆ which one obtains from the Main Theorem
The following proposition is central to the construction of an embedding of the order complex of non-crossing partitions into a spherical building
Summary
For every finite Coxeter group W of rank n, the order complex of the non-crossing partitions |NC(W )| is isomorphic to a chamber subcomplex of a spherical building ∆ of type An−1. This subcomplex is the union of a collection of apartments and has the homotopy type of a wedge of spheres. This result relies on the embedding of the Hurwitz graph into the chamber graph Γ∆ of the building ∆ which one obtains from the Main Theorem. The reader not familiar with buildings may want to read that section first
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