# Non-crossing Partition Lattices in Finite Real Reflection Groups

Document Type Article

Transactions of the American Mathematical Society, Vol. 360, no.4, pages 1983-2005

#### Abstract

For a finite real reflection group $W$ with Coxeter element $\gamma$ we give a case-free proof that the closed interval, $[I, \gamma]$, forms a lattice in the partial order on $W$ 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 simplicial generalised associahedron, and use this fact to give a new proof that the geometric realisation of this associahedron is a sphere.