Clebsch-Gordan Matrix Generalization I am trying to obtain the Clebsch-Gordan matrix that changes from the coupled angular momentum basis to the decoupled basis when coupling several $\frac{1}{2}$ spins.
So far, I have obtained the matrix for three spins by brute force; adding the two first spins and then adding the third one and obtaining their respective Clebsch-Gordan coefficients.
But for more spins doing it by brute force becomes quite tedious since the matrix size grows exponentially. I was wondering if there is a way to generalize the matrix expression for an arbitrary number $n$ of $\frac{1}{2}$ spins and if so, how can it be done.
Thanks in advance
 A: There is no easy way to generalize this.  In part this is because the basis states are not unique.  In the case of 3 spin-$1/2$ particles, there are two sets of $S=1/2$ states and any linear combination of $\vert 1/2,1/2\rangle_1$ and
$\vert 1/2,1/2\rangle_2$ is also a legitimate basis state.
The problem of multiplicities gets worse as you increase the number of spin-1/2 constituents. The number of times the final spin $S$ occurs in an $n$ spin-1/2 system is given in terms of dimensions of Young diagrams with $n$ boxes, and closely tied to the permutation of $n$ objects.  This is well understood but constructing states is usually done algorithmically using a computer.
The "more" canonical approach is precisely to use the permutation group.  In particular, using (for instance) class operators it's possible to construct different sets, but doing this manually is a serious pain.  There's also an approach based on Young projection operators, but it also becomes rapidly impractical to do this "by hand".
You can check out

Tung, Wu-Ki. Group theory in physics. Vol. 1. World Scientific, 1985

or

Ping, J., Wang, F., & Chen, J. Q. (2002). Group representation theory for physicists. World Scientific Publishing Company

or even the venerable

Hamermesh, M. (1989). Group theory and its applications to physical problems
for a sense of the techniques based on permutation groups.

