= Solution
<Branching of a representation> describes the <irreducible representation> content of a <restriction of a representation>:
$$
R\big|_H\cong\bigoplus_\lambda m_\lambda S_\lambda.
$$
This direct decomposition applies when the <restriction of a representation> is a <semisimple representation>, as for <finite groups> or continuous finite-dimensional <representations> of <compact groups>. For an arbitrary <group> and <subgroup>, complete reducibility need not hold. A <vector subspace> preserved by every $R(h)$ for $h\in H$ need not be preserved by the additional $R(g)$ for $g\notin H$, so an <irreducible representation> can become reducible.
For a concrete example, take the defining <representation> of $G=SU(2)$ on $\mathbb C^2$ and $H=\{\operatorname{diag}(z,z^{-1}):|z|=1\}$, a <subgroup> isomorphic to the <circle group>. The defining <representation> is <irreducible>: $SU(2)$ acts transitively on unit vectors, so no complex line is preserved by all its elements. Under $H$, the two coordinate lines are <invariant subspaces>, and
$$
\boxed{\mathbf2\big|_{U(1)}=\mathbf1_{+1}\oplus\mathbf1_{-1}.}
$$
Here the two <one-dimensional representations> act by $z$ and $z^{-1}$.
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