# Embed SU(2) into SO(6) or SU(4)

Here is a pattern of group embedding, $$SU(4) \supset SU(3) \times U(1)$$ such that the irrep of SU(4) can be decomposed as the sum of tensor product of irrep of SU(3) and irrep of U(1).

Two questions:

• I wonder what will be the embedding for SO(6), $$SO(6) \supset SU(2) \times G,$$ where, such that the irrep of SO(6) can be decomposed as the sum of tensor product of irrep of SU(2) and irrep of $$G$$?

• alternatively what will be the embedding for $$Spin(6)\simeq SU(4)$$, $$Spin(6)\simeq SU(4) \supset SU(2) \times G',$$ where, such that the irrep of SU(4) can be decomposed as the sum of tensor product of irrep of SU(2) and irrep of $$G'$$?

I only ask the proper ways to write the group $$G$$ and $$G'$$ here.

Here $$G$$ and $$G'$$ may be a product of several groups.

• you want to find all the possible $G$ and $G'$ on the right hand sides? – wonderich Oct 1 '18 at 4:57