Séminaire Lotharingien de Combinatoire, 93B.108 (2025), 12 pp.
Grant T. Barkley
Extended Weak Order for S~n and the Lattice of Torsion Classes
Abstract.
The extended weak order is a combinatorial poset associated to a Coxeter group, defined in terms of biclosed sets of roots in a root system. The lattice of torsion classes is an algebraic poset, defined in terms of sets of modules for an algebra. We show that the extended weak order for the affine symmetric group
Sn~ is in fact a lattice quotient of the lattice of torsion classes for the preprojective algebra of a cycle quiver. We show how this allows one to translate between algebraic and combinatorial perspectives. In particular, we show that the extended weak order on Sn~ encodes the exchange graphs of cluster algebras of type A~ via its lattice quotients.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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