A linear form φ on the Grothendieck group of an algebra determines a abelian, extension-closed subcategory of its finite length modules: the φ-semistable subcategory (in the sense of King). This subcategory is abelian and extension-closed. As φ varies, the subcategories picked out exhibit a wall-and-chamber structure. If the algebra is hereditary and finite type, we recover the combinatorics of Igusa-Orr-Weyman-Todorov pictures, or, equivalently, of the cluster complex. It turns out that for finite-type preprojective algebras, we obtain combinatorics described by Nathan Reading’s ‘shards’ (originally introduced by Reading to study the combinatorics of weak order on the associated Coxeter group). Shards provide a beautiful picture from which we can recover the combinatorics for any quotient of the preprojective algebra, including the hereditary cases. Time permitting, I will also say something about affine type. This project is joint work with David Speyer, and also draws on previous joint work with Osamu Iyama, Nathan Reading, and Idun Reiten.
This video was produced by Syracuse University Department of Mathematics as part of ICRA 2016.
