Past exam of the groups and representations course of the University of Oxford / 2021 / 4 / c / Solution

An unbroken subgroup $H$ must fix the vacuum expectation value itself. Thus a nonzero $H$-invariant vacuum expectation value can exist only if the restriction of a representation to $H$ contains a trivial representation. Applying the singlet criterion for an unbroken subgroup to solution givesThe two “no” entries are forbidden for every nonzero vacuum expectation value. Each “yes” means that the indicated subgroup can remain unbroken, subject to the scalar potential choosing such a vacuum. It does not by itself prove that the indicated subgroup is the entire stabilizer.

For exact spontaneous symmetry breaking to the stated subgroup, the distinction matters. A complex $4$ admits an exact $H_3$ stabilizer; a complex $6$ admits an exact $H_2$ stabilizer. A single $15$ preserving $H_2$ or $H_3$ necessarily preserves an additional continuous $U(1)$, as computed in solution, so neither exact endpoint is possible with that single field. Also, if the $6$ scalar field is constrained to the real structure of the SU4 exterior square, its nonzero stabilizer is larger than $H_2$. The paper does not specify whether this reality constraint is imposed, so both cases are distinguished below.