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.
@article{10_37236_7200,
author = {Julia Heller and Petra Schwer},
title = {Generalized non-crossing partitions and buildings},
journal = {The electronic journal of combinatorics},
year = {2018},
volume = {25},
number = {1},
doi = {10.37236/7200},
zbl = {1380.05198},
url = {http://geodesic.mathdoc.fr/articles/10.37236/7200/}
}
TY - JOUR
AU - Julia Heller
AU - Petra Schwer
TI - Generalized non-crossing partitions and buildings
JO - The electronic journal of combinatorics
PY - 2018
VL - 25
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/7200/
DO - 10.37236/7200
ID - 10_37236_7200
ER -
%0 Journal Article
%A Julia Heller
%A Petra Schwer
%T Generalized non-crossing partitions and buildings
%J The electronic journal of combinatorics
%D 2018
%V 25
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/7200/
%R 10.37236/7200
%F 10_37236_7200
Julia Heller; Petra Schwer. Generalized non-crossing partitions and buildings. The electronic journal of combinatorics, Tome 25 (2018) no. 1. doi: 10.37236/7200
