The CAGE (Combinatorics, Algebra, and Geometry) Seminar: Grant Barkley, 26 September 2024
Beyond weak order on affine Weyl groups
Grant Barkley | 2024-09-26 at 3:30 PM, at Korman Center (Drexel), Room 245
Abstract:
The ``poset of quasiregions'' in the affine braid arrangement, also known as the extended weak order on the affine symmetric group, is a partial order on certain total orders of the integers, or equivalently, on certain sets of positive roots in the type \(\widetilde{A}\) root system. It contains the weak Bruhat order on the affine symmetric group, which is the subposet on regions contained in the Tits cone. The extended weak order for a general Coxeter group is the subject of many conjectures. We will discuss the resolution of several of these conjectures in type \(\widetilde{A}\). In particular, we show that the extended weak order is a completely semidistributive lattice. Furthermore, we show how the poset of quasiregions encodes information about the type \(\widetilde{A}\) preprojective algebra and about type \(D\) cluster algebras. We also describe progress for other affine Weyl groups.
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