# Tensor product decomposition of SU(2)

I have a rather trivial question. I am looking for the decomposition of $1/2\otimes 1/2\otimes 1/2$. It should give, $0,1/2$ and $3/2$. I thought one must get as the overall dimension of this space 8, but counting, I just get 7. Does one have 2 singlets?

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Possible duplicate: Adding 3 electron spins. –  Emilio Pisanty Aug 31 '12 at 13:50
The question is similar, true but I am more interested in the formal mathematical expression of this problem. –  Hamurabi Aug 31 '12 at 15:19

In your particular case, repeated use of $$1/2 \otimes s = (s-1/2) \oplus (s+1/2)$$ gives $$1/2 \otimes 1/2 \otimes 1/2=(0\oplus 1) \otimes 1/2 = (0 \otimes 1/2) \oplus (1 \otimes 1/2) =1/2 \oplus (1/2 \oplus 3/2).$$ Thus one gets two spin 1/2 and one spin 3/2 representations.