## Non-crossing partition lattices in finite real reflection groups.(English)Zbl 1187.20051

Summary: 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.

### MSC:

 20F55 Reflection and Coxeter groups (group-theoretic aspects) 05E15 Combinatorial aspects of groups and algebras (MSC2010) 05A18 Partitions of sets 06A07 Combinatorics of partially ordered sets
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### References:

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