Séminaire Lotharingien de Combinatoire, 86B.54 (2022), 12 pp.
Grant T. Barkley and David E. Speyer
Biclosed Sets in Affine Root Systems
Abstract.
The extended weak order is a partial order associated to a Coxeter system (W,S). It is the containment order on ``biclosed'' sets of positive roots in the (real) root system associated to W. When W is finite, this order coincides with the weak order on W, and is a lattice; when W is infinite, the weak order on W is a proper order ideal in the extended weak order. It is a longstanding conjecture of Matthew Dyer that the extended weak order is a lattice for any W. We prove this conjecture for the affine Coxeter groups. Furthermore, we give a parametrization of the extended weak order for these groups in terms of the face lattice of the Coxeter arrangement for the associated finite group.
Received: November 25, 2021.
Accepted: March 4, 2022.
Final version: April 1, 2022.
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