Differential symmetric signature in high dimension

We study the differential symmetric signature, an invariant of rings of finite type over a field, introduced in a previous work by the authors in an attempt to find a characteristic-free analogue of the F-signature. We compute the differential symmetric signature for invariant rings k[x1, . . ., xn]G, where G is a finite small subgroup of GL(n, k), and for hypersurface rings k[x1, . . ., xn]/(f) of dimension ≥ 3 with an isolated singularity. In the first case, we obtain the value 1/|G|, which coincides with the F-signature and generalizes a previous result of the authors for the two-dimensional case. In the second case, following an argument by Bruns, we obtain the value 0, providing an example of a ring where differential symmetric signature and F-signature are different.