Moduli of $G$-constellations and crepant resolutions I: the abelian case

For a finite abelian subgroup $G\subset S{L}_n(\mathbb{C})$, we study whether a given crepant resolution $X$ of the quotient variety ${\mathbb{C}}^n\mathrm{/}G$ is obtained as a moduli space of $G$-constellations. In particular we show that, if $X$ admits a natural $G$-constellation family in the sense of Logvinenko over it with all fibers being indecomposable as $\mathbb{C}[{\mathbb{C}}^n]$-modules, then $X$ is isomorphic to the normalization of a fine moduli space of $G$-constellations.