that our hypersurface $X_A$ is quasi-smooth. We remark that Condition (1) is used once again when we introduce the BHK mirror in Section 2.2: it ensures that the matrix $A^T$ is a matrix of exponents of a polynomial with $n+1$ monomials and $n+1$ variables.

Group of Diagonal Automorphisms
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Let us discuss the groups of symmetries of the Calabi-Yau variety $X_{F_A}$. Firstly, consider the scaling automorphisms of ${\mathbb{C}}^{n+1} \setminus \{0\}$, $n\geq 2$. There is a subgroup, $({\mathbb{C}}^*)^{n+1}$, of the automorphisms of ${\mathbb{C}}^{n+1} \setminus \{0\}$. Explicitly, an element $ (\lambda_0,\ldots, \lambda_n) \in ({\mathbb{C}}^*)^{n+1}$ acts on any element $\mathbf{x} = (x_0,\ldots, x_n) \in {\mathbb{C}}^{n+1} \setminus \{0\}$ by: $$(\lambda_0,\ldots, \lambda_n) \times (x_0,\ldots, x_n)  \longmapsto  (\lambda_0x_0,\ldots, \lambda_n x_n).$$ We view the weighted projective $n$-space $W{\mathbb{P}}^{n}(q_0,\ldots, q_n)$ as a quotient of ${\mathbb{C}}^{n+1}\setminus \{0\}$ by a subgroup ${\mathbb{C}}^* \subset ({\mathbb{C}}^*)^{n+1}$ consisting of the elements that can be written $(\lambda^{q_0/d}, \ldots, \lambda^{q_n/d}) $ for some $\lambda \in {\mathbb{C}}^*$.

Moreover, there is a second subgroup of $({\mathbb{C}}^*)^{n+1}$, denoted ${\text{Aut}}(F_A)$, which can be defined as $${\text{Aut}}(F_A) := \{ (\lambda_0, \ldots, \lambda_n) \in ({\mathbb{C}}^*)^{n+1} | F_A(\lambda_0 x_0, \ldots, \lambda_n x_n) = F_A(x_0, \ldots, x_n)\text{ for all } (x_0,\ldots, x_n)\}.$$ This group is sometimes referred to as the group of diagonal 