Simple objects in torsion-free classes over preprojective algebras of Dynkin type

In this talk, I propose to study exact-categorical structures of torsion(-free) classes of module categories. For functorially finite torsion-free class, indecomposable projective and injective objects are easily described by \(\tau^-\)-tilting modules, and in particular, the numbers of them coincide. However, there can be more simple objects in torsion-free class, which I propose to study. I explain that the number of simple objects controls the validity of the Jordan–Hölder type theorem in a torsion-free class.

Then I’ll talk about simple objects in a torsion-free class over the preprojective algebra (and path algebra) of Dynkin type, which is also important in Lie theory due to Geiss–Leclerc–Schröer’s categorification of the cluster structure. By Mizuno’s result, we can associate an element \(w\) of the Weyl group to each torsion-free class \(\mathcal{F}\). By (extended) Gabriel’s theorem, \(\mathcal{F}\) roughly corresponds to the inversion set of \(w\), the set of positive roots which are sent to negative by \(w^{-1}\). Then I show that simple objects in \(\mathcal{F}\) are in bijection with Bruhat inversions of \(w\), which are related to the Bruhat order of the Weyl group.