Generalized McKay quivers of rank three

TL;DR

This paper extends the McKay correspondence to rank three by introducing generalized Cartan matrices for finite subgroups of SL(3,C), analyzing their properties, and explicitly classifying the associated McKay quivers.

Contribution

It introduces generalized Cartan matrices for SL(3,C) subgroups, demonstrating their properties and providing a complete classification of McKay quivers in this setting.

Findings

Generalized Cartan matrices are positive semi-definite.

Complete classification of SL(3,C) McKay quivers.

Extension of McKay correspondence to rank three.

Abstract

For each finite subgroup G of SL(n, C), we introduce the generalized Cartan matrix C_{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 semi-definiteness as in the classical case of affine Cartan matrices (the case of SL(2,C)). The complete McKay quivers for SL(3,C) are explicitly described and classified based on representation theory.