# Generalized non-crossing Partitions and Buildings

## TL;DR

This paper demonstrates that the lattice of non-crossing partitions for any finite Coxeter group can be embedded into a spherical building, providing new insights into their structure and properties across different types.

## Contribution

It introduces a novel embedding of non-crossing partition lattices into spherical buildings and analyzes their supersolvability and Hurwitz graph properties across types.

## Key findings

- Embedding of NC(W) into spherical buildings for all finite Coxeter groups.
- Proof that NC(A_n) is supersolvable for all n.
- Lower bounds on the Hurwitz graph radius for all types.

## 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 ${n \choose 2}$.