This is the author's Ph.D. thesis. We introduce two related invariants for
local (and standard graded) rings called differential and syzygy symmetric
signature. These are defined by looking at the maximal free splitting of the
module of K\"ahler differentials and of the the top-dimensional syzygy module
of the residue field respectively. We study and compute them for different
classes of rings: two-dimensional ADE singularities, two-dimensional cyclic
singularities, and cones over plane elliptic curves (for the differential
symmetric signature). The values obtained coincide with the F-signature of such
rings in positive characteristic. The thesis contains also a short survey on
the Auslander correspondence for quotient singularities.Comment: The main results of the thesis appeared also in arXiv:1602.07184 and
arXiv:1603.0642
