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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Non-crossing partition lattices in finite real reflection groups


Authors: Thomas Brady and Colum Watt
Journal: Trans. Amer. Math. Soc. 360 (2008), 1983-2005
MSC (2000): Primary 20F55; Secondary 05E15
Published electronically: October 23, 2007
MathSciNet review: 2366971
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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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Additional Information

Thomas Brady
Affiliation: School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland
Email: tom.brady@dcu.ie

Colum Watt
Affiliation: School of Mathematical Sciences, Dublin Institute of Technology, Kevin St., Dublin 8, Ireland
Email: colum.watt@dit.ie

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04282-1
PII: S 0002-9947(07)04282-1
Received by editor(s): January 27, 2005
Received by editor(s) in revised form: December 17, 2005
Published electronically: October 23, 2007
Article copyright: © Copyright 2007 American Mathematical Society