Saito duality between Burnside rings for invertible polynomials

TL;DR

This paper introduces an equivariant Saito duality as a Fourier transform on Burnside rings, revealing a duality between monodromy zeta functions of Berglund-H"ubsch dual invertible polynomials based on their symmetry groups.

Contribution

It extends Saito duality to an equivariant setting and connects it to monodromy zeta functions of dual invertible polynomials, providing a new perspective on their symmetry relations.

Findings

Reduced equivariant monodromy zeta functions are Saito dual for dual polynomials.

The duality explains relations between geometric roots of zeta functions.

The duality generalizes previous results on polynomial pairs.

Abstract

We give an equivariant version of the Saito duality which can be regarded as a Fourier transformation on Burnside rings. We show that (appropriately defined) reduced equivariant monodromy zeta functions of Berglund-H\"ubsch dual invertible polynomials are Saito dual to each other with respect to their groups of diagonal symmetries. Moreover we show that the relation between "geometric roots" of the monodromy zeta functions for some pairs of Berglund-H\"ubsch dual invertible polynomials described in a previous paper is a particular case of this duality.