Equivariant Poincare series of filtrations

TL;DR

This paper introduces a novel equivariant Poincaré series as an element of the Grothendieck ring, providing explicit computations for filtrations on plane germs with finite group actions.

Contribution

It proposes a new definition of equivariant Poincaré series as an element of the Grothendieck ring, extending previous power series approaches.

Findings

Defined an equivariant Poincaré series in the Grothendieck ring

Computed the series for filtrations on plane germs with finite group actions

Provided explicit examples and calculations

Abstract

We offer a new approach to a definition of an equivariant version of the Poincar\'e series. This Poincar\'e series is defined not as a power series, but as an element of the Grothendieck ring of $G$-sets with an additional structure. We compute this Poincar\'e series for natural filtrations on the ring of germs of functions on the plane $(\C^2,0)$ with a finite group representation.