Mirrors
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We now introduce the Shioda map and relate it to BHK mirrors. Recall the hypersurfaces $X_A$ and $X_{A^T}$ as above. Define the matrix $B$ to be $ dA^{-1}$ where $d$ is a positive integer so that $B$ has only integer entries. The Shioda maps are the rational maps $$\begin{aligned}
\phi_B: {\mathbb{P}}^n \dashrightarrow W{\mathbb{P}}^n(q_0,\ldots, q_n),\text{ and } \\
\phi_{B^T}: {\mathbb{P}}^n \dashrightarrow W{\mathbb{P}}^n(r_0,\ldots, r_n), 
\end{aligned}$$ where $$\begin{aligned}
(y_0: \ldots:y_n) &\stackrel{\phi_B}{\mapsto} (x_0:\ldots: x_n), \quad x_j = \prod_{k=0}^n y_k^{b_{jk}}, \text{ and }\\
(y_0: \ldots:y_n) &\stackrel{\phi_{B^T}}{\mapsto} (X_0:\ldots: X_n), \quad X_j = \prod_{k=0}^n y_k^{b_{kj}}.
\end{aligned}$$

Consider the polynomial $$F_{dI} : = \sum_{i=0}^n y_i^d$$ and the Fermat hypersurface cut out by it, $X_{dI} := Z(F_{dI}) \subset {\mathbb{P}}^n$. Note that the Shioda maps above restrict to rational maps $X_{dI} \stackrel{\phi_B}{\dashrightarrow} X_A$ and $X_{dI} \stackrel{\phi_B}{\dashrightarrow} X_{A^T}$, respectively, allowing us to obtain the diagram: $$\xymatrix{ & X_{dI} \ar@{-->}[dl]_{\phi_B} \ar@{-->}[dr]^{\phi_{B^T}} & \\
        X_A && X_{A^T} }$$

We now reinterpret the groups $G$ and $G^T$ in the context of the Shioda map. Any element of ${\text{Aut}}(F_{dI})$ is of the form $g=(e^{2\pi i h_j/d})_j$, for some integers $h_j$. When we push forward the action of $g$ via $\phi_B$, we obtain the diagonal automorphism $$(\phi_B)_*(g) : = (e^{{\frac{2\pi i }{d} \sum_{j=0}^n b_{ij}h_j}})_i 