Symmetry breaking , subgroup embeddings and the Weyl group

We present a systematic approach to writing adjoint Higgs vacuum expectation values (vevs), which break a symmetry G to differently embedded isomorphic copies of a subgroup belonging to the chain G ⊃ H1 ⊃ · · · ⊃ Hl, as linear combinations of each other. Given an adjoint Higgs vacuum expectation value h breaking G → H, a full complement of vevs breaking G to different embeddings of the subgroup H can be generated through the Weyl group orbit of h. An explicit formula for recovering each vev is given. We focus on the case when H stabilizes the highest weight of the lowest dimensional fundamental representation, where the formula is exceedingly simple. We also discuss cases when the Higgs field is not in the adjoint representation and apply these techniques to current research problems, especially in domain-wall brane model building. Electronic address: [email protected] Electronic address: [email protected] Electronic address: [email protected] Electronic address: [email protected]