The relativistic S-matrix with spin transforms as an invariant tensor under the action of the homogeneous Lorentz group. This transformation property enables one to analyze the structure of the S-matrix from the viewpoint of group theory. Here this procedure is generalized and applied to tensor fields which are invariant under one of the classical groups. In each case the number of linearly independent tensor fields on a given orbit is determined. The results may be useful in particle physics.