# Decomposition of the fundamental representation of $SU(2)\times SU(2)$

What is the decomposition of the 4-dimensional representation of $Spin(4)=SU(2)\times SU(2)$ under $SO(4)$? I need this to find $SO(4)$ singlets inside $Spin(4)$.

• What 4-dimensional representation are you talking about? Apr 11, 2017 at 21:45
• Any of them (I think there are 3?)
– Kosm
Apr 12, 2017 at 4:28
• If you know what the three are, then presumably you know how they decompose? One is the standard rep, the other two are trivial plus either left or right spinor rep. Apr 12, 2017 at 4:47
• No, I don't know the decomposition rules under subgroups. That's why I'm asking.
– Kosm
Apr 12, 2017 at 4:58
• If you know how they decompose into ${\rm Spin}(4)$ irreps then it suffices to just check if said irreps are also ${\rm SO}(4)$ irreps (which is essentially just seeing if they are ${\rm SO}(4)$ reps in the first place). Also, ${\rm SO}(4)$ is not a subgroup of ${\rm Spin}(4)$, it's a quotient group.. Apr 12, 2017 at 5:50