Non-crossing partition lattices in finite real reflection groups

Brady, Thomas; Watt, Colum

doi:10.1090/S0002-9947-07-04282-1

Non-crossing partition lattices in finite real reflection groups
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by Thomas Brady and Colum Watt PDF

Trans. Amer. Math. Soc. 360 (2008), 1983-2005 Request permission

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 we use this fact to give a new proof that the geometric realisation of this associahedron is a sphere.

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