How to get $3n-j$ symbols using the young tableau method?

Question

How do one compute the $3n-j$ symbols using the young tableau method,for a $U(n)$ ?

Edit

$3j$ symbols were first introduced by Wiegner which are related to C.G. coefficients, similarly $6j$ symbols are related to racah coefficients, similarly there are $9j$ symbols and so on, for short 3n-j symbols. They are useful in coupling.

There must be a way to calculate it using young tableau method or an algorithm?

In the 9th chapter of the book Group theory Book by Predrag Cvitanović, explains how to compute 3nj symbols using the birdtracks diagrams and young tableau, I had difficulty in understanding, how to compute it?

Answer 1

I don't think Young tableaux can be used to obtain the (generalized) Clebsh-Gordan coefficients or Wigner symbols. They contain much more information than what you can get out of the typical Young tableaux techniques. Young tableaux can help you to compute the multiplicities of the irreducible representations in a tensor product (Littlewood-Richardson rule). You can also use them to compute the dimension of the irreducible representation corresponding to a given Young tableaux that appears in the direct sum decomposition (hook length formula). The only thing one can do that goes slightly into the direction of what you asked is to obtain a projector to a specific irreducible subspace of a tensor product consisting only of factors of the defining representation of the group. Moreover, as far as I know, all of this applies directly only to $SU(n)$, not to $U(n)$, because $SU(n)$ has the same irreps as $SL(n,\mathbb C)$, which is the setting where Young tableaux naturally arise.

A standard reference would be the book by Fulton and Harris.