Flops and mutations for crepant resolutions of polyhedral singularities

Let $G$ be a polyhedral group $G\subset SO(3)$ of types $\mathbb{Z}\mathrm{/}n\mathbb{Z}$, $D_{2n}$ and $\mathbb{T}$. We prove that there exists a one-to-one correspondence between flops of $G$-Hilb${\mathbb{C}}^3$ and mutations of the McKay quiver with potential which do not mutate the trivial vertex. This correspondence provides two equivalent methods to construct every projective crepant resolution for the singularities ${\mathbb{C}}^3\mathrm{/}G$, which are constructed as moduli spaces ${\mathcal{M}}_C$ of quivers with potential for some chamber $C$ in the space $\mathrm{\Theta }$ of stability conditions. In addition, we study the relation between the exceptional locus in ${\mathcal{M}}_C$ with the corresponding quiver $Q_C$, and we describe explicitly the part of the chamber structure in $\mathrm{\Theta }$ where every such resolution can be found.