On McKay quivers and covering spaces

Abstract

In this paper, we study the relationship between the McKay quivers of finite subgroups of special linear groups and of general linear groups, via natural extension and embedding. We show that under certain condition, the McKay quiver of a finite subgroup G of GL( m ;C) is a regular covering of the McKay quiver of its normal subgroup G ∩ SL( m ;C), and when embedding G in a canonical way into SL( m + 1;C), the new McKay quiver is obtained by adding an arrow from the Nakayama translation of i back to i for each vertex i . We also show by examples that certain interesting McKay quivers can be obtained in these ways.