The Representation Theorem for Linear, Isotropic Tensor Functions Revisited

Abstract

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)=λ_3E_3 +λ_1E_1+λ_5E_5 for (nonzero) constants λ_3, λ_1 and λ_5. The noninvertible case results from considering C+λI.