# 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.

• I hope you have studied the numerous answers of this site on the ambiguities and quandaries of combining three spin 1/2s. The multiplicities of the reduced reps in combining an arbitrary number of spin 1/2s is straightforward: the Catalan triangle. But the enormous Clebsch matrix itself, is of course, not available. Commented Mar 7, 2022 at 14:58
• Related: physics.stackexchange.com/q/29443 and links therein Commented Mar 7, 2022 at 14:59
• Your results presumably agree with this? Commented Mar 7, 2022 at 15:08

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.