Speaker: Grant Barkley, Harvard University
Title: “Formal geometry of affine Coxeter arrangements”
Abstract:We study quasiregions of infinite Coxeter arrangements, which can be identified with biclosed subsets of the associated positive root system. They organize into a partial order which extends the weak Bruhat order on the group. Recently, we showed in joint work with David Speyer that this poset is a complete lattice when W is an affine Coxeter group, resolving a conjecture of Matthew Dyer for these groups. To do so, we introduce the notion of a clean arrangement, where every quasiregion is in fact a region. Focusing on the affine symmetric group (type A tilde), we show that the completely join-irreducible elements of this lattice correspond to shards of its Coxeter arrangement, as is known for finite Coxeter groups. We give a combinatorial description of the shards using cyclic non-crossing arc diagrams. We also show these objects biject with certain modules, called real bricks, over a type A tilde preprojective algebra, and discuss the relationship between the extended weak order and the lattice of torsion classes for this algebra.

The Discrete Math Seminar (DMS) is intended for Kennesaw State faculty working in the various areas of algebra, number theory, and discrete mathematics to get together to discuss their current work or related questions. Seminars often involve advanced mathematical knowledge. However, the seminars are open to anyone interested in attending. This talk will take place in-person and virtually.