PBW parametrizations and generalized preprojective algebras

Gei ss-Leclerc-Schroer (2017) [24] has introduced a notion of generalized preprojective algebras associated with generalized Cartan matrices and their symmetrizers. These algebras realize crystal structures on the set of maximal dimensional irreducible components of the nilpotent varieties (Geiss et al. 2018, [26]). For general finite types, we give stratifications of these components via partial orders of torsion classes in module categories of generalized preprojective algebras in terms of Weyl groups. In addition, we realize Mirkovic-Vilonen poly topes from generic modules of these components, and give an identification as crystals between the set of Mirkovic-Vilonen polytopes and the set of maximal dimensional irreducible components. This generalizes results of Baumann-Kamnitzer (2012) [8] and Baumann-Kamnitzer-Tingley (2014) [10].(c) 2021 Elsevier Inc. All rights reserved.