\text{ is $G$-invariant}\right\} \subseteq {\text{Aut}}(F_{A^T}).$$

If the group $G$ is a subgroup of $SL(F_A)$, then the dual group $G^T$ contains the group $J_{F_{A^T}}$.

It is sufficient to show that the element $\prod_{j=0}^n \rho_j^T$ is in the dual group $G^T$. This is equivalent to $\prod_{j=0}^n x_j$ to be $G$-invariant. Any element $(\lambda_0, \ldots, \lambda_n)$ of $G$ acts on the monomial $\prod_{j=0}^n x_j$ by $\prod_{j=0}^n \lambda_j = 1$ (as $G\subseteq SL(F_A)$).

If the group $G$ contains $J_{F_A}$, then the dual group $G^T$ is contained in $SL(F_{A^T})$.

The proof of this lemma is analogous to the lemma above. As the dual group $G^T$ sits between $J_{F_{A^T}}$ and $SL(F_{A^T})$, define the group $\tilde G^T : = G^T/ J_{F_{A^T}}$. We have a well-defined Calabi-Yau orbifold $Z_{A^T, G^T} := X_{A^T}/ \tilde G^T \subset W{\mathbb{P}}^n(r_0, \ldots, r_n)/\tilde G^T$. The Calabi-Yau orbifold $Z_{A^T, G^T}$ is the BHK mirror to $Z_{A,G}$.

Classical Mirror Symmetry for BHK Mirrors
-----------------------------------------

In this section, we summarize some results of Chiodo and Ruan for BHK mirrors. This section is based on Section 3.2 of [@CR]. We recommend the exposition there. Recall that we can view the weighted projective $n$-space $W{\mathbb{P}}^n(q_0,\ldots, q_n)$ as a stack $$\left[ {\mathbb{C}}^{n+1} \setminus \{0\} / {\mathbb{C}}^*\right]$$ where a group 