Computing resolutions of quotient singularities

Let G⊆GL(n) be a finite group without pseudo-reflections. We present an algorithm to compute and verify a candidate for the Cox ring of a resolution X→Cn/G, which is based just on the geometry of the singularity Cn/G, without further knowledge of its resolutions. We explain the use of our implementation of the algorithms in Singular. As an application, we determine the Cox rings of resolutions X→C3/G for all G⊆GL(3) with the aforementioned property and of order |G|≤12. We also provide examples in dimension 4.