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$\DeclareMathOperator\SO{SO}$Consider the $7$-dimensional $\mathbb R^7$ real irreducible orthogonal representation of $\SO(3)$. I am seeking a description of the orbit space (when the action is restricted to the sphere) that is as complete as possible. What is it isometric to? What are the orbit types? etc...

It would be even better if there is a nice description of orbits spaces for all real irreps of $\SO(3)$. Any help or reference would be appreciated.

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    $\begingroup$ I'm confused about the terminology being used here. When you say orbit space, what is the space being acted on? A (unitary) irrep of SO(3) means that this group is acting irreducibly on some C^n by isometries; are you referring to the action on the unit sphere of V? $\endgroup$
    – Yemon Choi
    Commented Aug 22, 2022 at 16:17
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    $\begingroup$ I edited the question. Does that help? $\endgroup$
    – miniii
    Commented Aug 22, 2022 at 16:26
  • $\begingroup$ If I'm correct, there are closed 5-dimensional subsets $X_2,X_3$ in $\mathbf{R}^7$ (hence 4-dimensional in the 6-sphere) consisting of points whose stabilizer is cyclic of order 2 (for $X_2$) and of order 3 (for $X_3$), except for points of $X_2\cap X_3$, which has dimension 3 and consists of points that have a stabilizer that is a conjugate of $\mathrm{SO}(2)$. Points outside $X_2\cup X_3$ have a trivial stabilizer. $\endgroup$
    – YCor
    Commented Aug 22, 2022 at 16:34
  • $\begingroup$ A reference or elaboration on this would be great. In arxiv.org/pdf/1109.1739.pdf Table 1, it is shown that the action has no boundary in the orbit space where the boundary is the closure of codimension-$1$ strata. Wouldn't a stabilizer $\text{SO}(2)$ yield a codimension $1$ stratum? $\endgroup$
    – miniii
    Commented Aug 22, 2022 at 16:54
  • $\begingroup$ Paper referenced in the comment: Gorodski and Lytchak - On orbit spaces of representations of compact Lie groups. $\endgroup$
    – LSpice
    Commented Aug 22, 2022 at 17:09

1 Answer 1

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I don't know where the orbit types in this case were first explicitly classified, but it is done in my paper Second order families of special Lagrangian 3-folds, Perspectives in Riemannian geometry, 63–98, CRM Proc. Lecture Notes 40, Amer. Math. Soc., Providence, RI, 2006. MR2237106. See Proposition 1 of Section 3. I include the zero, but it's easy to remove that and see what all the nonzero orbit types are.

Because the ring of $\mathrm{SO}(3)$-invariant polynomial invariants on the irreducible representation of dimension $7$ is generated in degrees $2$, $4$, $6$, $10$, and $15$ (with the square of the $15$-degree polynomial expressible as a polynomial in the lower degree ones), you actually have to go quite a distance before you can distinguish all of the orbits. The possible stabilizer types are $\mathrm{SO}(2)$, $A_4$, $S_3$, $\mathbb{Z}_3$, $\mathbb{Z}_2$, and $\{1\}$.

However, it's not hard to distinguish the $\mathbb{Z}_2$-quotient that you get by dividing by the action of $\mathrm{O}(3)$.

One way to describe the $\mathrm{O}(3)$-orbits on $S^6\subset\mathbb{R}^7$ is as un-ordered triples of points on the $2$-sphere modulo the rotations.

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  • $\begingroup$ Thank you for your answer! The proposition seems very helpful. I will be looking into it. $\endgroup$
    – miniii
    Commented Aug 22, 2022 at 18:24

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