Non-crossing partition lattices in finite real reflection groups

Abstract

For a finite real reflection group W W with Coxeter element γ \gamma we give a case-free proof that the closed interval, [ I , γ ] [I, \gamma ] , forms a lattice in the partial order on W W induced by reflection length. Key to this is the construction of an isomorphic lattice of spherical simplicial complexes. We also prove that the greatest element in this latter lattice embeds in the type W W simplicial generalised associahedron, and we use this fact to give a new proof that the geometric realisation of this associahedron is a sphere.