On weakly isotropic tensors

We define and characterize weakly isotropic tensors in terms of properties of invariance with respect to some sets of rotations, for instance, those in the cubic group. Hence, in particular, we characterize weakly isotropic tensors which are isotropic or skew-isotropic, depending on whether their order is even or odd, respectively. Restricting our attention to tensors of order less than or equal to 7, we characterize a minimal set of rotations such that invariance with respect to it implies the property of weak isotropy. Then we show that any weakly isotropic tensor of order 7 can be represented in terms of 36 independent arbitrary scalars. This number of independent scalars is minimal, in the sense that any representation for such a tensor involves at least 36 scalars. We produce two such minimal representations, first by finding a suitable choice of independent components in terms of which we express the remaining tensor components. Second, we find 36 linearly independent weakly isotropic tensors which are products of a Ricci symbol with two Kronecker symbols and are such that any other weakly isotropic tensor can be expressed as a linear combination of them