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

  • 1
    $\begingroup$ 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. $\endgroup$ Commented Mar 7, 2022 at 14:58
  • $\begingroup$ Related: physics.stackexchange.com/q/29443 and links therein $\endgroup$ Commented Mar 7, 2022 at 14:59
  • $\begingroup$ Your results presumably agree with this? $\endgroup$ Commented Mar 7, 2022 at 15:08

1 Answer 1


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


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.