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BHK mirror symmetry as introduced by Berglund--H\"ubsch and Marc Krawitzbetween Landau--Ginzburg (LG) models has been the topic of much study in recentyears. An LG model is determined by a potential function and a group ofsymmetries. BHK mirror symmetry is only valid when the group of symmetries iscomprised of the so-called diagonal symmetries. Recently, an extension to BHKmirror symmetry to include nonabelian symmetry groups has been conjectured. Inthis article, we provide a mirror map at the level of state spaces between theLG A-model state space and the LG B-model state space for the mirror modelpredicted by the BHK mirror symmetry extension for nonabelian LG models. Weintroduce two technical conditions, the Diagonal Scaling Condition, and theEquivariant $\Phi$ condition, under which a bi-degree preserving isomorphism ofstate spaces (the mirror map) is guaranteed to exist, and we prove that thecondition is always satisfied if the permutation part of the group is cyclic ofprime order.

Mirror Map for Landau-Ginzburg models with nonabelian groups