Relations between grand unified and monopole theories

Two kinds of interrelationships between GUTs and monopole theories are discussed: how the duality conjectures could have a bearing on understanding GUTs, and how some of the mathematical technology used in monopole studies can yield simple (Dynkin) diagrammatic rules for some of the common GUT group theory calculations. A compact notation for semisimple Lie algebras is supplied by Dynkin diagrams. Minimal fundamental weights are seen to define minimal representations into which matter may be placed, and also define a special direction for the adjoint Higgs field. Minimal weights play a special role, therefore, both in defining matter multiplets and in symmetry breaking. After considering gauge groups G broken down to U(1) X K/Z (with K semisimple) by an adjoint representation (AR) Higgs, it is asked how the representations of G will look when decomposed into irreducible representations of U(1) X K, by proving two theorems as given. The point is pedagogical: using concepts like the Weyl group, practical calculations can be performed with simple Dynkin diagrams