The Representation Theorem for Linear, Isotropic Tensor Functions Revisited

Two recent notes 1–2] express ideas that can be combined in order to get an elementary proof of the representation theorem of the title, a proof stressing the geometrical action of 4-tensors in subspaces. Thus we examine direct consequences, for linear maps C of Lin into Lin, of the notion of invariance (we recall this concept and we recall other standard notations in the first section). Suppose we classify the invariant (under Orth) subspaces of Lin, that an invariant linear C maps invariant subspaces into invariant subspaces and that Lin = Skw Sph Dev expresses the unique decomposition of Lin into a direct sum of three-, one- and five-dimensional invariant subspaces. Then, if we assume that C is also invertible, the structure of C comes from the knowledge of the invariant linear maps in Skw and Dev. As they must be, in each case, multiples of the identity, CE=C(E _{_3}+E _{_1}+ E _{_5})=_{_3} E _{_3} +_{_1} E _{_1}+_{_5} E _{_5} for (nonzero) constants _{_3}, _{_1} and _{_5}. The noninvertible case results from considering C+I.