Mirror pairs of quintic orbifolds

Two constructions of mirror pairs of Calabi-Yau manifolds are compared by example of quintic orbifolds $\mathcal{Q}$ . The first, Berglund–Hubsch–Krawitz, construction is as follows. If $X$ is the factor of the hypersurface $\mathcal{Q}$ by a certain subgroup $H'$ of the maximum allowed group $SL$, the mirror manifold $Y$ is defined as the factor by the dual subgroup ${H'}^{T}$. In the second, Batyrev, construction, the toric manifold $T$ containing the mirror $Y$ as a hypersurface specified by zeros of the polynomial $W_Y$ is determined from the properties of the polynomial $W_X$ specifying the Calabi-Yau manifold $X$. The polynomial $W_Y$ is determined in an explicit form. The group of symmetry of the polynomial $W_Y$ is found from its form and it is tested whether it coincides with that predicted by the Berglund–Hubsch–Krawitz construction.