Noncrossing partitions and Bruhat order

Abstract

We give a criterion for Bruhat order on noncrossing partitions corresponding to the Coxeter element $c=s_1{s}_2\cdots s_n$. Using it we prove that the Bruhat order endows noncrossing partitions with a lattice structure. We then explain what happens if we change the Coxeter element; in that case the lattice property fails and we explain which order to consider to get the same lattice structure as for the Coxeter element $c$. In particular we get bijections between noncrossing partitions associated to distinct Coxeter elements, which fix the set of reflections.

This paper has been withdrawn by the author since it gave rise to a joint work with Nathan Williams, see http://arxiv.org/abs/1503.00595