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

We study GIT quotients X_θ=V\!/\!\!/\!_θ G whose linearisation map defines an isomorphism between the group of characters of G and the Picard group of X_θ modulo torsion. Our main result establishes that the Cox ring of X_θ is isomorphic to the semi-invariant ring of the θ-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.