  We study first order deformations of the tangent sheaf of resolutions of
Calabi-Yau threefolds that are of the form $\mathbb{C}^3/Z_r$, focusing on the
cases where the orbifold has an isolated singularity. We prove a lower bound on
the number of deformations for any crepant resolution of this orbifold. We show
that this lower bound is achieved when the resolution used is the G-Hilbert
scheme, and note that this lower bound can be found using methods from string
theory. These methods lead us to a new way to construct the G-Hilbert scheme
using the singlet count.
