Representation-theoretical characterization of canonical custodial symmetry in NHDM potentials

By considering the basis-covariant constituents of N-Higgs-doublet potentials, we derive necessary and sufficient conditions for canonical $\mathrm{SO}{\left(4\right)}_C$ Custodial Symmetry (CS) of potentials with $N>2$ doublets, based on representation-theoretical and geometrical relations. In essence, our characterization relates the presence of canonical CS to basis-covariant vectors corresponding to particular bases of the defining representation of the orthogonal Lie algebras. For $N=3,4$ and 5, the conditions demand little computational effort to be evaluated, and we provide practical algorithms that may be efficiently implemented in a computer program, for deciding whether or not a potential is custodial-symmetric.