In wikipedia the following statement is made,
"the tensor product representation decomposes as the direct sum of one copy of each of the irreducible representations of dimension $2J+1$, where $J$ takes values from $|j_1-j_2|$ to $j_1+j_2$."
I don't understand the statement. For as much as I understand, for the system we consider two types of Hilbert spaces:
The Hilbert space which is spanned by the eigenvectors of the uncoupled system, and these are tensor products of the eigenstates of each angular momentum.
The Hilbert space which is spanned by the eigenvectors of the coupled system, which are primarly eigenstates of the total angular momentum.
And I know that in order to transition from the uncoupled basis to the coupled, each ket in the coupled basis is expressed as a linear combination basis kets in the uncoupled basis, multiplied with a corresponding Clebsch-Gordan coefficient.
But the statement I do not understand. In fact the statement looks to me as if it is saying the reverse, meaning:
the tensor product representation i.e an eigenstate of the uncoupled Hilbert space such as $|j_1,j_2;m_1,m_2\rangle$ is decomposed (expressed or expanded) as a sum of the coupled basis kets corresponding to a value $J$. Which is the total opposite of what the Clebsch-Gordan formula does.