Stable absolute minima of Higgs potentials with high rank representations

We present a simple method for constructing absolute minima of a Higgs potential built out of a single irreducible representation and which contains at least one cubic invariant. The stability properties of the isotropy groups of these minima under small perturbations of all Higgs parameters are investigated. As an example, the method is applied to the SU(5) model with the 75 underscore to generate a new class of stable and absolute minima. The limitations of the method are also discussed particularly in connection with an amended version of Michel's conjecture. (orig.)