# 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?
– anon
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.
– anon
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..
– anon
Apr 12, 2017 at 5:50