I am working with a SU(2) invariant system which has symmetry generators of the form $$ \lambda_1 = \begin{pmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 \end{pmatrix}\;;\qquad \lambda_2 = \begin{pmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & -i & 0\\ 0 & i & 0 & 0\\ 0 & 0 & 0 & 0 \end{pmatrix}\;;\qquad \lambda_3 = \begin{pmatrix} 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 \end{pmatrix} $$ It can be checked easily that the generators satisfy the su(2) algebra $[\lambda_i,\lambda_j] = 2i \epsilon_{ijk}\lambda_k$, where the $\epsilon_{ijk}$ is the completely antisymmetric Levi-Civita tensor.
If we now define unitary matrices of the form $U = e^{i\sum_i \alpha_i \lambda_i}$, the matrices will be elements of the SU(2) group. I have now the following questions:
- Is this an irreducible representation of the SU(2) group with dimension 4, or is this reducible?
- If it is a reducible representation, what will be the decomposition of the representation in terms of irreducible representations? I am asking this because I want to find the characters for this representation.
P.S. I have a physics background. So, I only have a basic understanding of group theory and group representations.
Edit 1: I see now that the representation is reducible with help from the comments. The space spanned by the vectors (1,0,0,0) and (0,0,0,1) form an invariant subspace, thus implying that it is a reducible representation. Any help with the second question is still appreciated.