A McKay correspondence for the Poincar\'e series of some finite subgroups of ${\rm SL}_3(\CC)$

TL;DR

This paper extends the McKay correspondence to certain non-abelian finite subgroups of SL_3(C), relating their invariants to surface singularities obtained via hyperplane sections, thus generalizing known results from SL_2(C).

Contribution

It introduces a new relation between non-abelian subgroups of SL_3(C) and surface singularities, broadening the scope of the McKay correspondence beyond classical cases.

Findings

Establishes a relation between group invariants and surface singularities.

Shows hyperplane sections yield Kleinian or Fuchsian surface singularities.

Generalizes McKay correspondence to SL_3(C) subgroups.

Abstract

A finite subgroup of ${\rm SL}_2(\CC)$ defines a (Kleinian) rational surface singularity. The McKay correspondence yields a relation between the Poincar\'e series of the algebra of invariants of such a group and the characteristic polynomials of certain Coxeter elements determined by the corresponding singularity. Here we consider some non-abelian finite subgroups $G$ of ${\rm SL}_3(\CC)$. They define non-isolated three-dimensional Gorenstein quotient singularities. We consider suitable hyperplane sections of such singularities which are Kleinian or Fuchsian surface singularities. We show that we obtain a similar relation between the group $G$ and the corresponding surface singularity.