Mirror map for Fermat polynomials with a nonabelian group of symmetries

We study Landau–Ginzburg orbifolds $(f,G)$ with $f=x_1^n+\cdots+x_N^n$ and $G=S\ltimes G^d$, where $S\subseteq S_N$ and $G^d$ is either the maximal group of scalar symmetries of $f$ or the intersection of the maximal diagonal symmetries of $f$ with $SL_N(\mathbb{C})$. We construct a mirror map between the corresponding phase spaces and prove that it is an isomorphism restricted to a certain subspace of the phase space when $n=N$ is a prime number. When $S$ satisfies the parity condition of Ebeling–Gusein-Zade, this subspace coincides with the full space. We also show that two phase spaces are isomorphic for $n=N=5$.