Mirror symmetry of nonabelian Landau–Ginzburg orbifolds with loop type potentials

Abstract

We study the bigraded vector space structure of Landau–Ginzburg orbifolds with the permutation symmetries of variables. We calculate the formula of the Poincaré polynomial for a certain loop polynomial orbifoldized by a semidirect product of a diagonal symmetry group and cyclic permutation group and check that the generalization of Berglund–Hübsch transpose gives mirror symmetric pairs for these cases.