One mistake I made is that

because

but this application is wrong.

The main reason is that, the is “living” in a tensor product space, while is living in ordinary space.

We can also see that, the norm of left side is 1, but the norm of the right side is not.

Using the Clebsch-Gordon series, we can deduce the product of spherical harmonics.

First, we need to know the relationship between the Wigner D-matrix and spherical harmonics. Using the equation

We can set and

Thus,

Now, recall the Clebsch-Gordon series,

set

rename some labels

We can multiply both side by and sum over , using