# Compute SO(3) action orbits and stabilizers with representation theory

I am asked to compute the orbits and isotropy groups (a.k.a. stabilizers) of the Lie group action \begin{align*} \varphi\colon \operatorname{SO}(3)\times\operatorname{Sym}(3) & \rightarrow \operatorname{Sym}(3) \\ (R,A) &\mapsto R A R^t \end{align*} where $$\operatorname{Sym}(3)$$ is the space of symmetric $$3\times 3$$ matrices. If I did it well, for an element $$A\in\operatorname{Sym}(3)$$, its orbit $$\mathcal{O}(A)$$ and isotropy group $$G_A$$ are given by:

$$\begin{equation*} \mathcal{O}(A) = \{B\in\operatorname{Sym}(3) \ \big| \ \operatorname{Eigenvalues}(B)=\operatorname{Eigenvalues}(A) \} \end{equation*}$$

\begin{align*} G_A \cong \begin{cases} \operatorname{SO}(3) &\text{ if A has 3 identical eigenvalues}\\ \operatorname{O}(2) &\text{ if A has 2 identical eigenvalues}\\ F &\text{ if A has no identical eigenvalue} \end{cases} \end{align*} where $$F$$ is the finite subgroup $$\{I,\ \operatorname{diag}(1,-1,-1),\ \operatorname{diag}(-1,1,-1),\ \operatorname{diag}(-1,-1,1)\}$$ of $$\operatorname{SO}(3)$$.

What I did is mostly to use standard linear algebra results for the case of $$A$$ diagonal and then extend to the non-diagonal case using that:

• Any $$A$$ is always on the same orbit of a diagonal matrix with the same eigenvalues.
• For two elements on the same orbit, their isotropy groups are conjugated and, therefore, isomorphic.

Now I am curious about if there is a more "sophisticated" way of computing these orbits and stabilizers using some results or techniques about representation theory. I don't know much about it, but I believe that the following map is a representation of $$\operatorname{SO}(3)$$: \begin{align*} \Pi \colon \operatorname{SO}(3) &\rightarrow \operatorname{GL}(\operatorname{Sym}(3)) \\ R &\mapsto \varphi_R \end{align*} where $$\varphi_R(A) = \varphi(R,A) = R A R^t$$.

Thanks a lot for your help!

• Sometimes representation theory is just linear algebra. I think this is one of those cases.
– anon
May 15, 2020 at 19:04
• Sym(3) is isomorphic to the space of homogeneous polynomials in variable $x,y,z$ by taking $\mathbb{R}^3 \otimes \mathbb{R}^3$ to be the space of $3 \times 3$ real matrices. Then polynomials become symmetric matrices by the identification ($e_1,e_2,e_3$ standard basis) $xy \to e_1 \otimes e_2 + e_2 \otimes e_1$ and $x^2 \to 2 e_1 \otimes e_1$ etc... Here laplacian of polynomial=trace of matrix. So the standard 5d irrep of degree two homogeneous harmonic polynomials is identified with symmetric traceless matrices. But ultimately I agree with @runway44 your way is best to find orbits Jan 21, 2022 at 15:58