I'm looking at irreducible, real representations of $SO(3)$. The 5-dimensional irrep is isomorphic to the space of trace-free quadratic forms on $\mathbb{R}^3$, and we all know that any such quadratic form can be diagonalized by the standard $SO(3)$ action on $\mathbb{R}^3$. My question is: is there any analogous "normal form" result for the 7-dimensional irrep, regarded as the space of trace-free cubic forms on $\mathbb{R}^3$, under the standard action of $SO(3)$?


More generally, I'd like to better understand the geometry of the orbits of these actions. Can anyone recommend a reference?


-Jeanne



Normal form for trace-free real cubic forms in 3 variables under SO(3)-action?
One way to do this is to reduce this to understanding sets of points on the $2$-sphere up to a certain equivalence. If you let $\mathcal{H}\_k$ denote the homogeneous polynomials $p$ of degree $k$ on $\mathbb{R}^3$ that are harmonic, i.e., that satisfy $\Delta p = 0$, then $\mathcal{H}\_k$ is an irreducible $\mathrm{SO}(3)$-representation of dimension $2k{+}1$. The space $\mathcal{P}\_k$ of all homogeneous polynomials of degree $k$ can be written as the direct sum
$$
\mathcal{P}\_k = \mathcal{H}\_k \oplus (R\cdot\mathcal{H}\_{k-2})
\oplus (R^2\cdot\mathcal{H}\_{k-4}) \oplus \cdots 
$$
where $R = {x\_1}^2+{x\_2}^2+{x\_3}^2$. In particular, there is a canonical 'harmonic part' $H(p)\in \mathcal{H}\_k$ for any $p\in \mathcal{P}\_k$, and the map $H:\mathcal{P}\_k\to \mathcal{H}\_k$ is linear. Then one has the following `factorization result':


**Proposition:** Any $p\in \mathcal{H}\_k$ can be 'factored' in the form
$p = H(\ell\_1\ell\_2\cdots\ell\_k)$,
where the 'factors' $\ell\_i\in \mathcal{H}\_1$ satisfy $|\ell\_1| = |\ell\_2| = \cdots = |\ell\_k|$ and the elements $\ell\_i$ are unique up to permutation and replacement of an even number of them by their negatives.


This 'harmonic factorization' is $\mathrm{SO}(3)$-equivariant, so finding normal forms for harmonic polynomials of a fixed degree $k$ reduces to normalizing sets of $k$ points on the $2$-sphere up to permutation and replacing an even number of them by their negatives, which is a combinatorial problem.


The proof of the above proposition goes through the standard representation theory of $\mathrm{SU}(2)$, which is the double cover of $\mathrm{SO}(3)$.


