PBW parametrizations and generalized preprojective algebras
arXiv:2011.06524 · doi:10.1016/j.aim.2021.108144
Abstract
Geiß-Leclerc-Schröer [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 Mirković-Vilonen polytopes from generic modules of these components, and give an identification as crystals between the set of Mirković-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 Études Sci. 120 (2014)].
arXiv comment62 pages, v3: minor corrections, added explanations, updated references
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