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On the Weak Order of Coxeter Groups

Part of: Lie algebras and Lie superalgebras Lattices Special aspects of infinite or finite groups

Published online by Cambridge University Press:  10 January 2019

Matthew Dyer
Matthew Dyer*
Affiliation:
Department of Mathematics, 255 Hurley Building, University of Notre Dame, Notre Dame, Indiana 46556, USA Email: dyer.1@nd.edu
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Abstract

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This paper provides some evidence for conjectural relations between extensions of (right) weak order on Coxeter groups, closure operators on root systems, and Bruhat order. The conjecture focused upon here refines an earlier question as to whether the set of initial sections of reflection orders, ordered by inclusion, forms a complete lattice. Meet and join in weak order are described in terms of a suitable closure operator. Galois connections are defined from the power set of $W$ to itself, under which maximal subgroups of certain groupoids correspond to certain complete meet subsemilattices of weak order. An analogue of weak order for standard parabolic subsets of any rank of the root system is defined, reducing to the usual weak order in rank zero, and having some analogous properties in rank one (and conjecturally in general).

Keywords

Coxeter grouproot systemweak orderlattice

MSC classification

Primary: 20F55: Reflection and Coxeter groups
Secondary: 06B23: Complete lattices, completions 17B22: Root systems
Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 2 , April 2019 , pp. 299 - 336
Copyright
© Canadian Mathematical Society 2018 

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