ИСТИНА

The Berglund–Hübsch–Henningson (BHH–) duality is a duality on the set of pairs (f, G) consisting of an invertible polynomial group and a subgroup G of diagonal symmetries of f. Symmetries of invariants of BHH-dual pairs are related to the mirror symmetry. There is a method to extend the BBH-duality to the set of pairs (f, G'), where G' is the semidirect product of a group G of diagonal symmetries of f and a group S of permutations of the coordinates preserving f. The construction is based on ideas of A.Takahashi and therefore is called the Berglund-Hübsch-Henningson-Takahashi (BHHT-) duality. Invariants of BHHT-dual pairs have symmetries similar to mirror ones only under some restrictions on the group S: the so-called parity condition (PC). Under the PC-condition it is possible to prove symmetries of the orbifold Euler characteristic for actions on the Milnor fibers of dual pairs. Moreover, the proves permit to make a conjecture that dual pairs possess more fine symmetries. The talk is based on joint results with W.Ebeling.