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short explanation:

I am trying to leverage my limited understanding of representation theory to explain (to myself) how many non-vanshing components of, for example, nonlinear optical susceptibility tensor ##\chi^{(2)}_{\alpha\beta\gamma}## can one have in a crystal with known point symmetry group ##G##. As I understand, all I need to do is to find how many trivial irreps occur in the represntation of the ##G## over any type of tensor space I am interested in. Every trivial irrep allows for one non-vanishing component. Am I correct?

long explanation:

We shall consider only local optical response (i.e. dipole approximation), so there is no spatial dispersion, thefore the only thing that matters is the point group symmetry of the crystal. Lets call it ##G##. I know what ##G## is. I can also find its character table, and I can calculate the actual matricies (##\mathbf{r}\left(a\right):\mathbb{R}^3\to\mathbb{R}^3, a \in G##) that give its representation over the space of 3D vectors.

Next we consider the rank-3 tensors (##\chi^{(2)} \in \mathbb{V}=\mathbb{R}^3\otimes\mathbb{R}^3\otimes\mathbb{R}^3##). The represntation of ##G## is easily obtained from the direct product of matricies for vectors, i.e. ##\mathbf{r}_{\chi^{(2)}}(a\in G): \mathbb{V}\to\mathbb{V}##, such that ##\mathbf{r}_{\chi^{(2)}}(a)=\mathbf{r}(a)\otimes\mathbf{r}(a)\otimes\mathbf{r}(a)##. Lets say there is a component of the susceptibility tensor that we observe in the experiment ##\chi^{(2)}\in\mathbb{V}##. Clearly, it cannot change under symmetry transformations of the crystal, so ##\chi^{(2)}=\mathbf{r}_{\chi^{(2)}}(a)\cdot\chi^{(2)},\quad \forall a\in G##. But this means that ##\chi^{(2)}## lies in the sub-space of ##\mathbb{V}## over which the representation of ##G## is trivial. The character of trivial representation for each group element ##a\in G## is always ##\chi_{triv}\left(a\right)=1## (sorry for using ##\chi## to represent both the character and the suscetibility, both notations are ingrained).

It would seem therefore that the only way I can have non-zero entries in the susceptibility tensor is if the relevant representation contains trivial representation. This can be checked using traces and characters so

number of trivial irreps in ##\mathbf{r}_{\chi^{(2)}}## is ##\left(\chi_{triv},\mbox{Tr}\left(\mathbf{r}_{\chi^{(2)}}\right)\right)=\frac{1}{\#G}\sum_{a\in G} (1)\cdot\mbox{Tr}\left(\mathbf{r}_{\chi^{(2)}}(a)\right)=\frac{1}{\#G}\sum_{a\in G} \left[\mbox{Tr}\left(\mathbf{r}\left(a\right)\right)\right]^3##

and this is the number of non-zero nonlinear scusceptibility components allowed in the crystal, given its point symmetry. Is this correct?

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# A Nonlinear susceptibility and group reps

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