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?