Spherical harmonics functions are said to be "rotationally invariant"
On the Wikipedia page, it says:
In mathematics, a function defined on an inner product space is said to have rotational invariance if its value does not change when arbitrary rotations are applied to its argument. For example, the function $f(x,y) = x^2 + y^2$ is invariant under rotations of the plane around the origin.
However this is confusing. It sounds like "rotationally inert", where rotations basically have no effect. (Spinning the circle $x^2 + y^2=r^2$ around the z-axis doesn't change anything about the values of the function anywhere).
Here's what I understand:
SH are rotationally invariant, which $ROT_1( SH( g ) ) = SH( ROT_2( g ) )$, where $ROT_1$ is a SH-domain rotation and $ROT_2$ is a spatial domain rotation, where $ROT_1$ and $ROT_2$ produce the same resultant orientation.
I got this from page 18 of this paper
Am I right? What is the Wikipedia page talking about? Have I misunderstood that Wikipedia page?
Best Answer
A specific spherical harmonic is not rotationally invariant, and the Wikipedia article does not claim this. What is true is that the space of all spherical harmonics of a fixed degree is a finite-dimensional irreducible representation of the rotation group $\text{SO}(3)$, and that the spherical harmonics form particularly nice bases of these representations. (In particular, rotating a spherical harmonic gets you a linear combination of spherical harmonics.)