ИСТИНА

The Berglund–Hübsch–Henningson– (BHH–) duality is a duality on the set of pairs (f, G) consisting of an invertible polynomial f 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, Ĝ), where Ĝ 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 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 in some cases. Moreover, the proofs permit to make a conjecture that BHHT–dual pairs possess more fine symmetries. The talk is based on joint results with W.Ebeling.