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# On the weak order of Coxeter groups

Published:2018-05-21

• Matthew Dyer,
Department of Mathematics, 255 Hurley Building, University of Notre Dame, Notre Dame, Indiana 46556, USA
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## Abstract

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 group, root system, weak order, lattice
 MSC Classifications: 20F55 - Reflection and Coxeter groups [See also 22E40, 51F15] 06B23 - Complete lattices, completions 17B22 - Root systems

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