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

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