The symmetric signature

We define two related invariants for a d-dimensional local ring (R,,k) called syzygy and differential symmetric signature by looking at the maximal free splitting of reflexive symmetric powers of two modules: the top-dimensional syzygy module SyzdR(k) of the residue field and the module of Kähler differentials ΩR/k of R over k. We compute these invariants for two-dimensional ADE singularities obtaining 1/|G|, where |G| is the order of the acting group, and for cones over elliptic curves obtaining 0 for the differential symmetric signature. These values coincide with the F-signature of such rings in positive characteristic.