Past exam of the groups and representations course of the University of Oxford / 2020 / 3 / c / Solution

Under the stated SU(2) embedding, the first two components of the defining SU(3) representation form a doublet, while the third is fixed. Hence $3{\downarrow }_{SU(2)}=2\oplus 1$. Its dual representation restricts in the same way because the SU(2) doublet is self-dual.

For the Adjoint representation, restrict $8=3\otimes \overline{3}-1$. The SU(2) tensor product $2\otimes 2=3\oplus 1$ givesHere subtraction denotes removal of the trivial representation generated by the identity matrix; it is not a negative-dimensional summand. The first SU(3) factor is unchanged, so the requested branching of a representation rules areThe dimensions on the two sides are respectively $9=6+3$, $3=3$, $3=2+1$ and $8=3+2+2+1$.