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arXiv · 1712.07985

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

Wolfgang Ebeling
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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.

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Wolfgang Ebeling. 2017-12-21. A McKay correspondence for the Poincar\'e series of some finite subgroups of ${\rm SL}_3(\CC)$. https://arxiv.org/abs/1712.07985

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