of $SL(F_A)$ as a generator of $J_{F_A}$ is the element $(e^{2\pi i q_j/d})_j$ and $\prod_j e^{2\pi i q_j / d} = e^{\frac{2\pi i}{d} \sum_j q_j} = 1$. Fix a group $G$ so that $J_{F_A} \subseteq G \subseteq SL(F_A)$ and put $\tilde G := G/ J_{F_A}$. To help summarize, we have the following diagram of groups:

Consider the Calabi-Yau orbifold, $Z_{A,G} : = X_{F_A}/\tilde G\subset W{\mathbb{P}}^n(q_0, \ldots, q_n) / \tilde G$. We now will describe the Berglund-Hübsch-Krawitz mirror to it.

The Berglund-Hübsch-Krawitz Mirror
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In this section, we construct the BHK mirror to the Calabi-Yau orbifold $Z_{A,G}$ defined above. Take the polynomial $$F_{A^T} = \sum_{i=0}^n \prod_{j=0}^n X_j^{a_{ji}}.$$ It is quasihomogeneous because there exist positive integers $r_i :=\sum_j b_{ji}$ so that $$F_{A^T}(\lambda^{r_0} X_0 , \ldots, \lambda^{r_n} X_n) = \lambda^{d}F_{A^T}(X_0 , \ldots,  X_n).$$ Note that the polynomial $F_{A^T}$ cuts out a well-defined Calabi-Yau hypersurface $X_{A^T} \subseteq W{\mathbb{P}}^n(r_0, \ldots, r_n)$. Define the diagonal automorphism group, ${\text{Aut}}(F_{A^T})$, analogously to ${\text{Aut}}(F_A)$. By the analogous isomorphism to that in Equation \[autisom\], the group ${\text{Aut}}(F_{A^T})$ is generated by $\rho_i^T := {\text{diag}}(e^{2\pi i b_{ij}/d})_{j=0}^n \in ({\mathbb{C}}^*)^{n+1}$. Define the dual group $G^T$ relative to $G$ to be $$G^T : = \left\{\prod_{i=0}^n (\rho_i^T)^{s_i} \middle| s_i \in {\mathbb{Z}}, \text{ where }\prod_{i=0}^n x_i^{s_i} 