Hugh Thomas: Quotients of preprojective algebras and lattice quotients of weak order

Let B be Dynkin-type preprojective algebra. Mizuno showed that the lattice of torsion classes of B is isomorphic to weak order on the corresponding Weyl group W. We consider the lattice of torsion classes of algebra quotients B/I, and show that they are lattice quotients of weak order on W . I will describe some of the general setting, and then focus on type A, in which we can describe explicitly which lattice quotients arise in this way. We also show that these are exactly the simplicial quotients (i.e., those such that the corresponding coarsening of the Coxeter fan is simplicial). This gives a combinatorial criterion for when a lattice quotient of weak order in the symmetric group is simplicial; no combinatorial criterion was known previously. This work-in-progress is joint with Osamu Iyama, Nathan Reading, and Idun Reiten.