= Centralizer of the standard SU2 subgroup in SU3
{title2=$C_{SU(3)}(SU(2))=\{\operatorname{diag}(z,z,z^{-2}):|z|=1\}$}
Embed <SU(2)> into <SU(3)> by $U\mapsto\operatorname{diag}(U,1)$. Its defining <representation> decomposes as an irreducible doublet and a <trivial representation>. The <Schur lemma> forces every commuting <matrix> to be $\operatorname{diag}(zI_2,w)$. Unitarity and unit <determinant> imply $|z|=1$ and $w=z^{-2}$, proving the formula. With <Hermitian> generator $X=\operatorname{diag}(1,1,-2)$, the defining <representation> restricts as $\mathbf2_1\oplus\mathbf1_{-2}$ and the <Adjoint representation> as $\mathbf3_0\oplus\mathbf2_3\oplus\mathbf2_{-3}\oplus\mathbf1_0$. The <U(1) charges> in the <Adjoint representation> are differences of defining charges. The generated connected <subgroup> is $(SU(2)\times U(1))/\mathbb Z_2$, since the two factors meet in $\operatorname{diag}(-I_2,1)$.
Back to article page
