Flops and mutations for crepant resolutions of polyhedral singularities

Abstract: Let $G$ be a polyhedral group $G\subset SO(3)$ of types $\mathbb{Z}/n\mathbb{Z}$, $D_{2n}$ 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 $\mathbb{C}^3/G$, which are constructed as moduli spaces $\mathcal{M}_C$ of quivers with potential for some chamber $C$ in the space $\Theta$ 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 $\Theta$ where every such resolution can be found.