8th order isotropic tensor

Question

Does anyone knows what is the general form of an $8$th order isotropic tensor? $2$th order is $\delta_{ij}$, $4$th order it is $\lambda \delta_{ij} \delta_{kl}+\mu(\delta_{ik} \delta_{jl} + \delta_{il} \delta_{jk})$. what is the form of an $8$th order isotropic tensor? Thanks.

Answer 1

Constructing the complete set of isotropic 8th order tensors is messy - there are 105(!) fundamental tensors, of which only 91 are linearly independent.

There is a paper "Linearly Independent Sets of Isotropic Cartesian Tensors of Ranks up to Eight" which gives a procedure to construct them (free download from NIST) and also contains a "minimal" list.

The original reference for representations of isotropic tensors is Hermann Weyl's "The Classical Groups", 1939, Princeton Press.

Also see the responses to this question; someone had suggested a paper by Harold Jeffreys.

May I ask why you are interested in these representations (if it is not a problem)?

Answer 2

This is kind of very late to the party, but if anyone needs this stuff again, in this question

Finding basis of isotropic tensors of rank $n$

bases for isotropic tensors up to 8th-order have been computed using Mathematica, such that you can use them for explicit computations.