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.
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