PBW parametrizations and generalized preprojective algebras

TL;DR

This paper explores the structure of generalized preprojective algebras linked to Cartan matrices, revealing stratifications, crystal structures, and connections to Mirković-Vilonen polytopes, extending prior theoretical results.

Contribution

It introduces stratifications of irreducible components via torsion classes and links these to Mirković-Vilonen polytopes, generalizing previous work.

Findings

Stratifications of components via torsion classes

Realization of Mirković-Vilonen polytopes from modules

Identification of crystals with polytopes

Abstract

Gei\ss-Leclerc-Schr\"oer [Invent. Math. 209 (2017)] has introduced a notion of generalized preprojective algebra associated with a generalized Cartan matrix and its symmetrizer. This class of algebra realizes a crystal structure on the set of maximal dimensional irreducible components of the nilpotent variety [Selecta Math. (N.S.) 24 (2018)]. 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 Mirkovi\'c-Vilonen polytopes from generic modules of these components, and give an identification as crystals between the set of Mirkovi\'c-Vilonen polytopes and the set of maximal dimensional irreducible components. This generalizes results of Baumann-Kamnitzer [Represent. Theory 16 (2012)] and Baumann-Kamnitzer-Tingley [Publ. Math. Inst. Hautes \'Etudes Sci. 120 (2014)].