Séminaire Lotharingien de Combinatoire, 91B.105 (2024), 12 pp.
Grant T. Barkley
Shards for The Affine Symmetric Group
Abstract.
The poset of "biclosed sets" in a root system has received attention as a natural extension of the weak Bruhat order on the associated Coxeter group. We will discuss these ideas in the context of the simplest infinite Coxeter groups, the affine symmetric groups. Using the combinatorial model introduced in (Barkley and Speyer 2022), we show that many constructions for the weak order on the symmetric group have analogs for the extended weak order on the affine symmetric group. In particular, shards in the affine braid arrangement biject with completely join-irreducible elements of the extended weak order, and there is a parametrization of both objects by "type-A~ arc diagrams".
Received: November 15, 2023.
Accepted: February 15, 2024.
Final version: April 1, 2024.
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