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For a symmetrizable GCM C and its symmetrizer D, Geiss-Leclerc-Schr¨oer [Invent. Math. 209 (2017)] has introduced a generalized preprojective algebra Π associated to C and D, that contains a class of modules, called locally free modules. We show that any basic support τ- tilting Π-module is locally free and gives a classification theorem of torsion-free classes in repΠ as the generalization of the work of Mizuno [Math. Z. 277 (2014)].

We propose a definition of deformed symmetrizable generalized Cartan matrices with several deformation parameters, which admit a categorical interpretation by graded modules over the generalized preprojective algebras in the sense of Gei\ss-Leclerc-Schr\"oer. Using the categorical interpretation, we deduce a combinatorial formula for the inverses of our deformed Cartan matrices in terms of braid group actions. Under a certain condition, which is satisfied in all the symmetric cases or in all the finite and affine cases, our definition coincides with that of the mass-deformed Cartan matrices introduced by Kimura-Pestun in their study of quiver $\mathcal{W}$-algebras., Comment: 27 pages

早大学位記番号:新8777

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)]., Comment: 62 pages, v3: minor corrections, added explanations, updated references

For a symmetrizable GCM $C$ and its symmetrizer $D$, Geiss-Leclerc-Schr\"oer [Invent. Math. 209 (2017)] has introduced a generalized preprojective algebra $\Pi$ associated to $C$ and $D$, that contains a class of modules, called locally free modules. We show that any basic support $\tau$-tilting $\Pi$-module is locally free and gives a classification theorem of torsion-free classes in $\operatorname{\mathbf{rep}}{\Pi}$ as the generalization of the work of Mizuno [Math. Z. 277 (2014)]., Comment: 17 pages; v3: final version, slightly changed the title