Orbifold E-functions of dual invertible polynomials

An invertible polynomial is a weighted homogeneous polynomial with the number of monomials coinciding with the number of variables and such that the weights of the variables and the quasi-degree are well defined. In the framework of the search for mirror symmetric orbifold Landau–Ginzburg models, P. Berglund and M. Henningson considered a pair (f,G)(f,G) consisting of an invertible polynomial ff and an abelian group GG of its symmetries together with a dual pair (f˜,G˜). We consider the so-called orbifold E-function of such a pair (f,G)(f,G) which is a generating function for the exponents of the monodromy action on an orbifold version of the mixed Hodge structure on the Milnor fibre of ff. We prove that the orbifold E-functions of Berglund–Henningson dual pairs coincide up to a sign depending on the number of variables and a simple change of variables. The proof is based on a relation between monomials (say, elements of a monomial basis of the Milnor algebra of an invertible polynomial) and elements of the whole symmetry group of the dual polynomial.