Clebsch-Gordan coefficients, spin networks and intertwiners After spending some time with LQG books and articles i have still some problems regarding concepts of this theory.
Spin network is built from lines labeled by spin label $j$ and since angular momentum must be conserved at the node one needs to associate intertwiner with the node. What's the role of the Clebsch-Gordan coefficients? Aren't they precisely intertwiners asserting that momentas on each nodes are conserved?
 A: Clebsh-Gordan coefficients don't play a part in the definition of LQG. The notion of intertwining tensors is enough to build spin network states.
They are, however, a very convenient tool for practical calculations.
This has to do with representation theory of $\mathfrak{su}_2$. A spin-$j$ irrep can be thought of as the totally symmetric part of a tensor product of $2j$ fundamental (spin-$1/2$) irreps.
You can also implement intertwiners between $V_{\{1,2,3\}}$ as tensors acting on the tensor product $V_{1/2}^{\otimes 2 j_1} \otimes V_{1/2}^{\otimes 2 j_2} \otimes V_{1/2}^{\otimes 2 j_3}$, and later taking only the projection of these tensors on the symmetric subspace. This is convenient, because there exists a very simple graphical calculation model for these – see Penrose's binor calculus.
Clebsh-Gordan coefficients are essentially the ratios between the normalization of intertwiners in Penrose's binor calculus, and the renormalization of intertwiners in LQG (given by the spin network inner product).
They can be used, for example, to easily calculate the values of the $6j$ symbol that enters the definition of the Ponzano-Regge model (quantum gravity in $3d$ Euclidean space-time). You replace normalized intertwiners with symmetrized binor calculus diagrams which makes their evaluation trivial, and multiply by the Clebsh-Gordan coefficients to get the correct answer.
