Generalized McKay quivers of rank three

For each finite subgroup G of SL n (.), we introduce the generalized Cartan matrix A G in view of McKay correspondence from the fusion rule of its natural representation. Using group theory, we show that the generalized Cartan matrices have similar favorable properties such as positive semidefiniteness as in the classical case of affine Cartan matrices. The complete McKay quivers for SL 3(.) are explicitly described and classified based on representation theory.