Kashiwara-Saito realized crystal bases of quantum enveloping algebras on irreducible components of varieties of nilpotent modules over preprojective algebras for simply-laced types. Recently, Gei\ss-Leclerc-Schr\“oer generalized these realizations to non-simply laced types by developing representation theory of a class of $1$-Iwanaga-Gorenstein algebras and their preprojective algebras associated with symmetrizable GCMs (=generalized Cartan matrices) and their symmetrizers. In this talk, we relate representation theory of the generalized preprojective algebras with numerical data about the dual canonical bases, so called Lusztig data, for symmetrizable GCMs of finite types. In particular, we realize Mirkovi\’c-Vilonen polytopes from some generic modules over generalized preprojective algebras as a generalization of the work of Baumann-Kamnitzer-Tingley.

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zoom URL: https://kyoto-u-edu.zoom.us/j/84523537818
passcode: $E_7$型Weyl群の位数