I am wondering how to prove the following statement (which is widely used, for example in turbulence theory) mathematically rigorously:

Assume we are talking about $V=\mathbb{R}^3$. Given a tensor field $T_{i_1,i_2,...,i_k}$ of rank $k$, that depends on a vector $r_j \in V$. If we require that $T=A_{i_1} B_{i_2}...K_{i_k}T_{i_1,i_2,...,i_k}$ is invariant under O(3) for all vectors $A, B,...,K\in V$. Then its most general form involves the Kronecker Delta and the vector Specifically, for $k=2$, $$T_{ij} (r_s)= A(r) r_i r_j + B(r) \delta_{ij},$$ where $r$ is the Euclidean norm of $r_j$. If $k=3$, $$T_{ijl} (r_s)= A(r) r_i r_j r_l + B(r) r_i \delta_{jl} + C(r) r_j \delta_{il} + D(r) r_l \delta_{ij}$$ And so on and so forth. This basically amounts to saying that the only way such expression as T can be invariant under O(3) is if it is composed of pairs of scalar products. How does one prove this?

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    $\begingroup$ For starters, this falls under the subject of invariant theory, which can be approached using representation theory. $\endgroup$ – Qiaochu Yuan Jan 12 '18 at 20:45
  • $\begingroup$ Any textbook you can recommend on invariant theory? $\endgroup$ – Cyclone Jan 13 '18 at 10:19

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