Coinvariant algebras of finite subgroups of SL(3, C)

Abstract

For most of the finite subgroups of SL(3, C) we give explicit formulae for the Molien series of the coinvariant algebras, generalizing MCKay's formulae [MCKay99] for subgroups of SU(2). We also study the G-orbit Hilbert scheme HilbG(C3) for any finite subgroup G of SO(3), which is known to be a minimal (crepant) resolution of the orbit space C3/G. In this case the fiber over the origin of the Hilbert-Chow morphism from HilbG(C3) to C3/G consists of finitely many smooth rational curves, whose planar dual graph is identified with a certain subgraph of the representation graph of G. This is an SO(3) version of the MCKay correspondence in the SU(2) case.