Poincaré series and monodromy of a two-dimensional quasihomogeneous hypersurface singularity

Abstract

A relation is proved between the Poincaré series of the coordinate algebra of a two-dimensional quasihomogeneous isolated hypersurface singularity and the characteristic polynomial of its monodromy operator. For a Kleinian singularity not of type $A_{2n}$, this amounts to the statement that the Poincaré series is the quotient of the characteristic polynomial of the Coxeter element by the characteristic polynomial of the affine Coxeter element of the corresponding root system. We show that this result also follows from the McKay correspondence.

arXiv commentLaTeX2e, 12 pages

Cited by in corpus (4)

• McKay correspondence for the Poincaré series of Kleinian and Fuchsian singularities
• Kostant's generating functions, Ebeling's theorem and McKay's observation relating the Poincare series
• Poincaré series of relative symmetric invariants for SL${}_n(\mathbb{C})$
• Poincaré series and monodromy of the simple and unimodal boundary singularities