In this article, Table 1 lists different symmetry groups and the tensors that describe them. However, it is unclear how the authors reached these results. They just refer to the book, and it is unclear how they deduced the tensors relative to each symmetry group.
Could anyone explain to me the first example of the list? It says that the group $C_{\infty v}$ is described by the rank-3 tensor $T_1$ with components $T_{1}^{ijk}=\sqrt{5/2}\left[ n^in^jn^k-\frac{1}{5}\left( \delta^{ij}n^k+\delta^{jk}n^i+\delta^{ki}n^j\right)\right]$, where the unity vector $\hat{n}$ defines a rotation axis of the group $C_{\infty v}$. It presents other 6 rank-3 tensors, but why they aren't used is unclear.
Which are the rank-3 tensors that represents the symmetry group $C_{\infty v}$?
I know that the seven rank-3 tensors come from the multipolar expansion but my poor knowledge in general downs permit to understand everything. One of these tensors is the $T_2$ with components $T_2^{ijk}=\frac{1}{2}\left( m^im^jm^k-m^il^jl^k-m^jl^kl^i-m^kl^il^j\right)$, where $\hat{m}\times\hat{l}=\hat{n}$. The article says a third tensor $T_3$ together with $T_2$ describes the symmetry $D_{3h}$. The reason is unclear to me.
I know a bit about rank-n tensors but almost nothing about group theory. So, I'm sorry for my naive question.