Singularities with Symmetries, orbifold Frobenius algebras and Mirror Symmetry

Previously, we introduced a duality transformation for Euler $G$–Frobenius algebras. Using this transformation, we prove that the simple $A,D,E$ singularities and Pham singularities of coprime powers are mirror self–dual where the mirror duality is implemented by orbifolding with respect to the symmetry group generated by the grading operator and dualizing. We furthermore calculate orbifolds and duals to other $G$–Frobenius algebras which relate different $G$–Frobenius algebras for singularities. In particular, using orbifolding and the duality transformation we provide a mirror pairs for the simple boundary singularities $B_n$ and $F_4$. Lastly, we relate our constructions to $r$ spin–curves, classical singularity theory and foldings of Dynkin diagrams.