# Dual Invertible Polynomials with Permutation Symmetries and the Orbifold Euler Characteristic

## TL;DR

This paper proves that for cyclic permutation groups satisfying a specific parity condition, the reduced orbifold Euler characteristics of dual pairs of invertible polynomials coincide up to sign, extending mirror symmetry constructions.

## Contribution

It establishes a precise condition under which the orbifold Euler characteristics of dual pairs match, generalizing previous mirror symmetry frameworks.

## Key findings

- Orbifold Euler characteristics of dual pairs coincide up to sign.
- Cyclic permutation groups satisfying the parity condition are key.
- Extension of mirror symmetry construction to new symmetry groups.

## Abstract

P. Berglund, T. H\"ubsch, and M. Henningson proposed a method to construct mirror symmetric Calabi-Yau manifolds. They considered a pair consisting of an invertible polynomial and of a finite (abelian) group of its diagonal symmetries together with a dual pair. A. Takahashi suggested a method to generalize this construction to symmetry groups generated by some diagonal symmetries and some permutations of variables. In a previous paper, we explained that this construction should work only under a special condition on the permutation group called parity condition (PC). Here we prove that, if the permutation group is cyclic and satisfies PC, then the reduced orbifold Euler characteristics of the Milnor fibres of dual pairs coincide up to sign.