Generalized non-crossing partitions and buildings (Q1700789)

Summary: For any finite Coxeter group \(W\) of rank \(n\) we show that the order complex of the lattice of non-crossing partitions \(\text{NC}(W)\) 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 \(\binom{n}{2}\).