Equivariant Poincaré Series and Monodromy Zeta Functions of Quasihomogeneous Polynomials

Abstract

In earlier work, the authors described a relation between the Poincare series and the classical monodromy zeta function corresponding to a quasihomogeneous polynomial. Here we formulate an equivariant version of this relation in terms of the Burnside rings of finite abelian groups and their analogues. Let f(x1, . . . , xn) be a quasihomogeneous polynomial. In [1], [2], there was described a relation between the Poincare series PX(t) of the coordinate ring of a hypersurface singularity X = {f = 0} and the classical monodromy zeta function ζf(t) of the polynomial f . The relation involved the so called Saito duality: [7], [8]. Namely, in [2], it was shown that PX(t) ·OrX(t) = ζ ∗ f (t) . (1) Here OrX(t) is a rational function determined by the orbit types of the natural C-action on X (see, e.g., [3]), ζf = ζf(t)/(1 − t) is the reduced monodromy zeta function of f , and ζ f (t) is the Saito dual of ζf(t) with respect to the quasidegree of the polynomial f . This relation had no intrinsic explanation. It was obtained by computation of both sides and comparison of the results. In particular, the role of the Saito duality remained unclear. In [4], an equivariant version of the Saito duality for a finite abelian group was formulated as a transformation between the Burnside rings of the group G and of the group G of its characters. Here we use the Burnside rings and their analogues to define equivariant versions of ∗Partially supported by the DFG Mercator program (INST 187/490-1), the Russian government grant 11.G34.31.0005, RFBR–10-01-00678, NSh–8462.2010.1.