# Addition of $N$ spin halves

If I have two spin-halves, then \begin{align} \frac{1}{2} \otimes \frac{1}{2} = 0 \oplus 1. \end{align} If I have three spin-halves, then \begin{align} \frac{1}{2} \otimes \frac{1}{2} \otimes \frac{1}{2} = \frac{1}{2} \oplus \frac{1}{2} \oplus \frac{3}{2}, \end{align} and for four \begin{align} \frac{1}{2} \otimes\frac{1}{2} \otimes \frac{1}{2} \otimes \frac{1}{2} = 0 \oplus 0 \oplus 1 \oplus 1 \oplus 1 \oplus 2. \end{align}

What's the decomposition into irreducible representations for $N$ (say even) spin-1/2s? That is, how many spin-$0$s do I get, how many spin-$1$s, how many spin-$2$s etc... There will be only one spin-$N/2$.

Of course one can solve this by induction, but on the off-chance that someone knows what it is by heart?

• Is this the right formula? – Peter Shor Apr 24 '16 at 1:30
• Duplicate of 249635 – Cosmas Zachos May 1 '16 at 19:52
• – Cosmas Zachos May 1 '16 at 20:06