New Calabi–Yau orbifolds with mirror Hodge diamonds

We prove a representation-theoretic version of Borisov–Batyrev mirror symmetry, and use it to construct infinitely many new pairs of orbifolds with mirror Hodge diamonds, with respect to the usual Hodge structure on singular complex cohomology. We conjecture that the corresponding orbifold Hodge diamonds are also mirror. When XX is the Fermat quintic in P4P4, and X˜∗ is a Sym5Sym5-equivariant, toric resolution of its mirror X∗X∗, we deduce that for any subgroup ΓΓ of the alternating group A5A5, the ΓΓ-Hilbert schemes Γ-Hilb(X)Γ-Hilb(X) and Γ-Hilb(X˜∗) are smooth Calabi–Yau threefolds with (explicitly computed) mirror Hodge diamonds.