Does a faithful action of SO(3) imply a metric on R^3?

[mma]

I think that the usual action of SO(3) on R^3 (defined by matrix multiplication) is faithful, because to non-identity rotations belong non-identity transformations.If we don't have originally a norm on R^3, but do have a faithful action of SO(3) on it, then we can try to define a norm by taking the length of an arbitrarily selected nonzero vector as 1, and the length of all vectors in its orbit space also as 1 (so, we declared the sphere of radius 1 in R^3). We define the length of the multiples of these unit vectors as the absolute value of the multiplier. The question is that is this norm well-defined? For example, is it sure that the action of a 180 degree rotation will bring all vectors to its negative? Because if not, then this definition is evidently ambiguous. Or, what extra conditions must this action satisfy to make this norm-definition unambiguous?

 

[Ben Niehoff]

Actually, you can do a little better than that. Since SO(3) is defined as the group of transformations that leave a certain quadratic form invariant, then you can simply write down a generic 3-vector, write out the most generic quadratic form (it has 9 terms), act on it with SO(3) and see which terms are left invariant. This will determine the metric up to an overall scale.

This assumes that your vector transforms in the fundamental representation. The same procedure applies to any representation, though. Simply write out the most generic quadratic form, and see which terms remain invariant under the action of the group. For example, the symmetric, traceless tensor rep will have 5-tuples of numbers, and hence 25 terms in its quadratic form.

You can take a shortcut and note that the quadratic form must be diagonal, since SO(3)-invariance implies space is isotropic.

 

[Hurkyl]

I confess that when I read that, it looks more like it says "If you already know the representation theory of SO(3), this is how you would go about computing something" rather than something that says "Here is how you derive the representation theory of SO(3) from scratch".

 

[Ben Niehoff]

Yes. Isn't that what the OP was asking?

 

[mma]

Is every faithful action of SO(3) on R^3 linear? If yes, then my question is: why? if not, then my question is: do nonlinear faithful actions also imply a metric on R^3?

 

[mma]

Sorry, I'm afraid that I didn't understand you. Perhaps you talk about the general case, not only about the linear. But I don't know what is this "fundamental representation" and also not, why should a quadratic form in 3 dimensions have 25 components.