they showed that $X_A$ and $X_{A^T}/P$ satisfy the classical mirror symmetry relation in that $$h^{p,q} (X_A, {\mathbb{C}}) = h^{n-1-p, q} (X_{A^T}/P, {\mathbb{C}}).$$

This proposal fell out of favor when Batyrev and Borisov developed the powerful toric approach (see [@Ba], [@BB1], and [@BB2]). In the 2000s, Krawitz revived Berglund and Hübsch’s proposal by giving a rigorous mathematical description of their mirror and proving a mirror symmetry theorem on the level of Frobenius algebra structures [@Kr].

Krawitz also generalized the Berglund-Hübsch mirror proposal by introducing the notion of a dual group: We start with a polynomial $F_A$. Consider the group $SL(F_A)$ of phase symmetries of $F_A$ leaving $H^{n,0}(X_A)$ invariant. Define the subgroup $J_{F_A}$ of $SL(F_A)$ to be the group consisting of the phase symmetries induced by the ${\mathbb{C}}^*$ action on weighted-projective space (so that all elements of $J_{F_A}$ act trivially on the weighted-projective space). Take the group $G$ to be some subgroup of $SL(F_A)$ containing $J_{F_A}$, i.e., $J_{F_A} \subseteq G \subseteq SL(F_A)$. We obtain a Calabi-Yau orbifold $Z_{A,G} := X_A/ \tilde G$ where $\tilde G := G/J_{F_A}$. Consider the analogous groups $SL(F_{A^T})$ and $J_{F_{A^T}}$ for the polynomial $F_{A^T}$. Krawitz defined the dual group $G^T$ relative to $G$ so that $J_{F_{A^T}} \subseteq G^T\subseteq 