Can someone identify this vector representation of $\rm SO(3)$ in terms of multi-variable polynomials?

Question

I am trying to get some deeper intuition for the representations of $\rm SO(3)$ and how they combine with each other, and I ran into an odd object that I'm hoping that folks here might help me identify.

As a general setting, I am considering the combinations of the solid harmonics $$S_{\ell m}(\mathbf r) = r^\ell Y_{\ell m}(\theta,\phi)$$ (which are, conveniently, homogeneous polynomials of degree $\ell$, and are much easier to work with than the $Y_{\ell m}$'s), and I am trying to understands what actually happens, at the coordinate level, when you use Clebsch-Gordan coefficents to combine a set of these into a single representation. That is, I define $$ F_{\ell_1\ell_2\ell_3}(\mathbf r_1, \mathbf r_2, \mathbf r_3) = \sum_{m_1,m_2,m_3} ⟨\ell_1m_1\ell_2m_2|\ell_3m_3⟩ S_{\ell_1m_1}(\mathbf r_1) S_{\ell_2m_2}(\mathbf r_2) S_{\ell_3m_3}(\mathbf r_3)^* , $$ which naturally gives a scalar-valued function, and I then look to understand the resulting structure.

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As a first example, consider the case when $(\ell_1,\ell_2,\ell_3) = (1,1,1)$. It is an easy calculation (meaning: Mathematica can do it easily) to show that $$ F_{111}(\mathbf r_1, \mathbf r_2, \mathbf r_3) = c_{111} \mathbf r_1 \cdot (\mathbf r_2 \times \mathbf r_3), $$ where $c_{111}$ is a boring numerical constant. Similarly, if one takes the slightly more complex example of $(\ell_1,\ell_2,\ell_3) = (1,2,2)$, there is again a simple and pleasing structure: $$ F_{122}(\mathbf r_1, \mathbf r_2, \mathbf r_3) = c_{122} \mathbf r_1 \cdot (\mathbf r_2 \times \mathbf r_3)(\mathbf r_2 \cdot\mathbf r_3). $$

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My question here is what happens a couple of levels up, at $(\ell_1,\ell_2,\ell_3) = (2,2,3)$. Here the polynomials are no longer something that I can recognize, but Mathematica has very little trouble reducing the scalar function to the following form: $$ F_{223}(\mathbf r_1, \mathbf r_2, \mathbf r_3) = c_{223} \left(\mathbf r_1 \cdot (\mathbf r_2 \times \mathbf r_3)\right) \ \mathbf r_1 \cdot \begin{pmatrix} x_2 (y_3^2 + z_3^2 - 4 x_3^2) - 5 y_2 x_3 y_3 - 5 z_2 x_3 z_3 \\ y_2 (z_3^2 + x_3^2 - 4 y_3^2) - 5 z_2 y_3 z_3 - 5 x_2 x_3 y_3 \\ z_2 (x_3^2 + y_3^2 - 4 z_3^2) - 5 x_2 x_3 z_3 - 5 y_2 y_3 z_3 \end{pmatrix} . $$ And this brings me to my real question here: what is that thing on the right?

It is clear to me that, as a cubic-polynomial function of $\mathbf r_2$ and $\mathbf r_3$, this thing transforms as a vector and it is therefore some kind of $\ell=1$ representation of $\rm SO(3)$. But what representation is this, exactly? Does it have a name? Is there some systematic way to treat it? (Say, analogously to the formalism in this answer, maybe?) I have tried to get it as clean as I can, but I'm still not super sold on it, and it feels like there should be a better way to express it.

I would be particularly like to know how this object fits within a larger framework, as I would like to keep going to higher $\ell$'s (at least as high as $(\ell_1,\ell_2,\ell_3) = (2,3,4)$) and the more I can understand about the setting, the better.

Answer 1

OK, folks, I managed to get this on my own =).

Short answer: the vector in question can be expressed as $$ \begin{pmatrix} x_2 (y_3^2 + z_3^2 - 4 x_3^2) - 5 y_2 x_3 y_3 - 5 z_2 x_3 z_3 \\ y_2 (z_3^2 + x_3^2 - 4 y_3^2) - 5 z_2 y_3 z_3 - 5 x_2 x_3 y_3 \\ z_2 (x_3^2 + y_3^2 - 4 z_3^2) - 5 x_2 x_3 z_3 - 5 y_2 y_3 z_3 \end{pmatrix} = \mathbf r_2 r_3^2-5\mathbf r_3 (\mathbf r_2\cdot\mathbf r_3) $$ so that the target function reads $$ F_{223}(\mathbf r_1, \mathbf r_2, \mathbf r_3) = c_{223} \left(\mathbf r_1 \cdot (\mathbf r_2 \times \mathbf r_3)\right) \left[ (\mathbf r_1\cdot\mathbf r_2) r_3^2-5(\mathbf r_1\cdot\mathbf r_3)(\mathbf r_2\cdot\mathbf r_3) \right] . $$

Slightly longer answer.

How does one get to this?

For one, the vector representation is a bit of a red herring, and it is better to work directly with the scalar form (i.e. the big vector dotted with $\mathbf r_1$), as this allows for more freedom in collecting terms and recognizing patterns.

And secondly, the terms of the form $y_3^2 + z_3^2 - 4 x_3^2$ are the key things to look for, and these should be simplified as $$ y_3^2 + z_3^2 - 4 x_3^2 = r_3^2 - 5 x_3^2. $$ Once this separation is done, it is much easier to break those two terms into separate groups and simplify them individually.