Pleased to Meet you, Dr. Schur

It s easier to think of the group of rotations as acting on all of three-dimensional space, instead of just the sphere. Any point in space can be rotated about the origin the sphere is just an invariant of those rotations. Now here is the amazing mathematical revelation any group that acts on some space (such as three-dimensional space or the sphere), also acts on functions of that spatial variable. Here is an example of how it works. We are cowards, you and I, so we ll start with an easy example. 

You can check that this new action is linear. In other words, a group element acting on the sum of two functions will have the same result as it would were it to act on each function separately and then one were to sum them. In other words, if the space of functions is a vector space, which it is, then every single element of this group acts like a matrix on this space of functions. 

Ponder this. Hold this example in your head. Hold on tight to it because it is the inspiration for all that follows. 

For the case we care more about, the group SO(3, R) acts on functions in three space by rotating the spatial variable. So a function f(x, y, z) becomes f(Ax + By + Cz, Dx + Ey + Fz, Gx + Hy + Iz) where the matrix 

Let s introduce some simple notation for what is going on in these two examples. In both cases we have a group, say G, with group elements, one of which might be g. Each of these group elements acts on a function by changing the functional variables in some way. Let us denote that action as 

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