An equivariant Poincaré series of filtrations and monodromy zeta functions

Abstract

We define a new equivariant (with respect to a finite group $$G$$ action) version of the Poincare series of a multi-index filtration as an element of the power series ring $${\widetilde{A}}(G)[[t_1, \ldots , t_r]]$$ for a certain modification $${\widetilde{A}}(G)$$ of the Burnside ring of the group $$G$$ . We give a formula for this Poincare series of a collection of plane valuations in terms of a $$G$$ -resolution of the collection. We show that, for filtrations on the ring of germs of functions in two variables defined by the curve valuations corresponding to the irreducible components of a plane curve singularity defined by a $$G$$ -invariant function germ, in the majority of cases this equivariant Poincare series determines the corresponding equivariant monodromy zeta functions defined earlier.