Generalized non-crossing Partitions and Buildings

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 connected 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 $n$ and 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 $\left(\genfrac{}{}{0ex}{}{n}{2}\right)$.

arXiv commentminor corrections, 28 pages, 12 figures; keywords: Generalized non-crossing partitions, buildings, Hurwitz graph, supersolvability