The semi-invariant ring as the Cox ring of a GIT quotient

We study GIT quotients $X_{\theta }=V\text{\hspace{0.17em}⁣}\mathrm{/}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\mathrm{/}\text{\hspace{0.17em}⁣}{}_{\theta }G$ whose linearisation map defines an isomorphism between the group of characters of $G$ and the Picard group of $X_{\theta }$ modulo torsion. Our main result establishes that the Cox ring of $X_{\theta }$ is isomorphic to the semi-invariant ring of the $\theta$-stable locus in $V$. This applies to quiver flag varieties, Nakajima quiver varieties, hypertoric varieties, and crepant resolutions of threefold Gorenstein quotient singularities with fibre dimension at most one. As an application, we present a simple, explicit calculation of the Cox ring of the Hilbert scheme of $n$-points in the affine plane.