DIFFERENTIAL SYMMETRIC SIGNATURE IN HIGH DIMENSION

Abstract

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[x(1), ... , x(n)](G), where G is a finite small subgroup of GL(n, k), and for hypersurface rings k[x(1), ... , x(n)]/(f) of dimension >= 3 with an isolated singularity. In the first case, we obtain the value 1/vertical bar G vertical bar, 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.