I think that a reformulation of my question is necessary: An intertwiner $\iota:\; V_{j_{1}}\bigotimes V_{j_{2}}\rightarrow V_{j_{3}}$ is defined as:

$\forall g\in SU(2),\;\forall u_{i},v_{i},w_{i},...\in V_{j_{i}}:\;\iota((\rho_{j_{1}}\bigotimes\rho_{j_{2}})(g)\;[v_{1}\bigotimes v_{2}])=\rho_{j_{3}}(g)[\iota(v_{1}\bigotimes v_{2})]$

where $\rho_{j_{i}}$ is the representation map corresponding to the irrep of spin $j_i$ of $SU(2)$, and $V_{j_i}$ are the invariant spaces upon which acts the $\rho_{j_{i}}(g)$ for $g\in SU(2)$

I know that the Shur's lemma is: if

$\iota:\; V_{j}\rightarrow V_{k}$

is an interwtwiner, then is it either a scalar (if $j=k$) or zero ($j\not= k$)

Now, what I want to know, is if $V_{j_{1}}\bigotimes V_{j_{2}} = \dots\oplus V_{j_{2}} \oplus \dots$ take an example $V_{j_{1}}\bigotimes V_{j_{2}} = V_{j_{4}} \oplus V_{j_{2}} \oplus V_{j_{5}}$ I can write:

$\forall g\in SU(2),\;\forall u_{i},v_{i},w_{i},...\in V_{j_{i}}:\;\iota((\rho_{j_{4}}\oplus\rho_{j_{3}}\oplus\rho_{j_{5}})(g)\;[v_{1}\bigotimes v_{2}])=\rho_{j_{3}}(g)[\iota(v_{1}\bigotimes v_{2})]$

in this case how to prove that $\iota$ is a scalar? (by Shur's lemma)

`$W_1\otimes W_2 \to W_3$`

is a scalar, where the`$W_i$`

are irreps. But it will not be: indeed, unless`$W_1$`

is the trivial irrep, then`$W_1\otimes W_2$`

is not an irrep, and so certainly not isomorphic to`$W_3$`

, and so a map between them cannot be a multiple of the identity, because there is no identity between them. – Theo Johnson-Freyd May 16 '10 at 18:27