Let $ G $ be a cyclic group of order $ n $, acting on $ \mathbb{C}^n $ by the cyclic action $ (z_1, z_2, \ldots, z_n) \rightarrow (z_2, z_3, \ldots, z_1) $. Does the quotient $ \mathbb{C}^n / G $ (which is Spec of the cyclically invariant polynomials $ \mathbb{C}[z_1, \ldots, z_n]^G $) admit a crepant resolution?
For $ n=2 $, it is uninteresting. For $ n=3 $, the singularity can be computed to be $ \operatorname{Spec} \mathbb{C}[x,y,z,w]/(xy-z^3) $ and it's known that $ \operatorname{Spec} \mathbb{C}[x,y,z]/(xy-z^3) $ has a crepant resolution, see here for instance. I cannot compute anything reasonably beyond this, although I'm most interested in the case $ n=5 $.
