Singlet criterion for an unbroken subgroup ($V^H\ne \{0\}$)

An unbroken subgroup $H$ must fix the vacuum expectation value $v$ of a scalar field in a group representation $V$. Thus a nonzero such $v$ exists exactly when $V^H\ne \{0\}$. For a compact group, complete reducibility of compact-group representations identifies this fixed vector subspace with the sum of the trivial representations in the restriction of a representation $V{|}_H$. Every $v\in V^H$ has $H\subseteq \mathrm{Stab}(v)$; equality requires a separate stabilizer calculation. The criterion neither determines the full unbroken group nor guarantees that a given scalar potential has its minimum at that direction.