An equivariant Poincar\'e series of filtrations and monodromy zeta functions

TL;DR

This paper introduces an equivariant Poincaré series for filtrations with a finite group action, providing formulas and showing its relation to monodromy zeta functions for plane curve singularities.

Contribution

It defines a new equivariant Poincaré series in a modified Burnside ring and relates it to monodromy zeta functions for plane valuations and singularities.

Findings

Provides a formula for the equivariant Poincaré series using G-resolutions.

Shows the series often determines the equivariant monodromy zeta functions.

Extends classical invariants to an equivariant setting with group actions.

Abstract

We define a new equivariant (with respect to a finite group $G$ action) version of the Poincar\'e 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 Poincar\'e 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 Poincar\'e series determines the corresponding equivariant monodromy zeta functions defined earlier.