\sum_{j=1}^n a_j^\gamma.$$ We now may define the bigraded Chen-Ruan orbifold cohomology as a direct sum of twisted sector ordinary cohomology groups: $$H^{p,q}_{\text{CR}} ( [Z_{A,G}], {\mathbb{C}}) = \bigoplus_{\gamma \in G{\mathbb{C}}^*} H^{p-a(\gamma), q-a(\gamma)}( X_A^\gamma / G{\mathbb{C}}^*, {\mathbb{C}}).$$ The degree $d$ Chen-Ruan orbifold cohomology is defined to be the direct sum $$H^d_{\text{CR}} \left( [Z_{A,G}], {\mathbb{C}}\right)  = \bigoplus_{p+q=d} H^{p,q}_{\text{CR}} ( [Z_{A,G}], {\mathbb{C}}) .$$

Continue to assume that the group $G$ contains $J_{F_A}$ and is a subgroup of $SL(F_A)$ and the hypersurface $X_A$ is Calabi-Yau. Chiodo and Ruan prove:

Given the Calabi-Yau orbifold $Z_{A,G}$ and its BHK mirror $Z_{A^T, G^T}$ as above, one has the standard relationship between the Hodge diamonds of mirror pairs on the level of the Chen-Ruan cohomology of the orbifolds: $$H^{p,q}_{\text{CR}} ( [Z_{A,G}], {\mathbb{C}}) \cong H^{n-1-p,q}_{\text{CR}}([Z_{A^T,G^T}], {\mathbb{C}}).$$

This is a classical mirror symmetry theorem for such orbifolds. We remark that in the case of orbifolds the dimension of the bigraded Chen-Ruan orbifold cohomology vector spaces and stringy Hodge numbers agree. Moreover, we have:

Suppose both Calabi-Yau orbifolds $Z_{A,G}$ and $Z_{A^T,G^T}$ admit smooth crepant resolutions $M$ and $W$ respectively, then we have the equality $$h^{p,q}(M, {\mathbb{C}}) = h^{n-1-p,q}(W, {\mathbb{C}}),$$ where $h^{p,q}$ is the ordinary $(p,q)$ Hodge number.

The Shioda Map and BHK 