On the orbifold Euler characteristics of dual invertible polynomials with non-abelian symmetry groups
In the framework of constructing mirror symmetric pairs of Calabi-Yau manifolds, P.Berglund, T.Hubsch 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 (f~, 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 G~ coincide. From this we derive that the orbifold Euler characteristics of the Milnor fibres of dual periodic loop polynomials coincide up to sign.