Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 4 / 19I / b / Solution

Writing $\rho (A)X=A_{1}X{A}_1^{-1}$, we haveand $\rho (I)=I$, so this conjugation representation is indeed a group representation.

The underlying three-dimensional module isUsing $M_3(\mathbb{C})=\mathrm{End}(W)\cong W\otimes W^{\ast }$, the self-duality of finite-dimensional SU2 representations, and $V_1\otimes V_1\cong V_2\oplus V_0$, we obtainTherefore the irreducible decomposition is