The representation theorem for linear, isotropic tensor functions revisited (Q1976121)

Let \(V\) be a three-dimensional vector space, and let Lin denote a nine-dimensional vector space of linear maps from \(V\) to \(V\). The author shows that if \({\mathbf C}:\text{Lin} \to\text{Lin}\) is linear, invariant (under Orth) and invertible, then there exist scalars \(\mu,\alpha\), and \(\lambda \) such that \({\mathbf C}[{\mathbf H}] =\mu({\mathbf H}+ {\mathbf H}^T)+ \alpha ({\mathbf H}-{\mathbf H}^T)+ \lambda(tr {\mathbf H}) \mathbf{1}\) for every \({\mathbf H}\in \text{Lin}\). The proof is based on the decomposition formula \(\text{Lin}= \text{Skw} \oplus\text{Sph}\oplus \text{Dev}\), and on a study of linear operators in the subspaces Skw, Sph, and Dev. Here, Skw, Sph, and Dev denote the space of skew, spherical and deviatoric tensors, respectively.