In the framework of constructing mirror symmetric pairs of Calabi-Yau manifolds, P. Berglund, T. H\"ubsch and M. Henningson considered a pair $(f,G)$ consisting of an invertible polynomial $f$ and a finite abelian group $G$ of its diagonal symmetries and associated to this pair a dual pair $(\widetilde{f}, \widetilde{G})$. A. Takahashi suggested a generalization of this construction to pairs $(f, G)$ where $G$ is a non-abelian group generated by some diagonal symmetries and some permutations of variables. In a previous paper, the authors showed that some mirror symmetry phenomena appear only under a special condition on the action of the group $G$: a parity condition. Here we consider the orbifold Euler characteristic of the Milnor fibre of a pair $(f,G)$. We show that, for an abelian group $G$, the mirror symmetry of the orbifold Euler characteristics can be derived from the corresponding result about the equivariant Euler characteristics. For non-abelian symmetry groups we show that the orbifold Euler characteristics of certain extremal orbit spaces of the group $G$ and the dual group $\widetilde{G}$ coincide. From this we derive that the orbifold Euler characteristics of the Milnor fibres of dual periodic loop polynomials coincide up to sign.

中文翻译：

---

具有非阿贝尔对称群的对偶可逆多项式的轨道欧拉特征

Orbifold Euler 特征的镜像对称性可以从关于等变 Euler 特征的相应结果导出。对于非阿贝尔对称群，我们证明群$G$ 和对偶群$\widetilde{G}$ 的某些极值轨道空间的orbifold Euler 特征是重合的。由此我们推导出双周期环多项式的米尔诺纤维的轨道欧拉特性在符号上一致。