Algebraic aspects of broken symmetry: Irreducible representations of SO(3)

The steady-state configurations of physical systems with symmetry are determined as minima of invariant energy functions. The problem of finding these critical points becomes formidable if the carrier space is endowed with any but the simplest of group action. Nevertheless, such complicated conditions prevail in such diverse areas as symmetry breaking in elementary particle physics, phase changes in early universe models and bifurcations due to hydrodynamic instabilities. It is known that special elements of the configuration space exist (those whose little group is maximal) that are natural solutions to the equations that determine the extrema of the functional. For the most part this fact has been applied only piecemeal, using special algebraic properties of each case. In order to illuminate issues common to these problems it is worthwhile to consider the problem in its general, group theoretical, setting. We have considered the problem of minimizing the most general invariant polynomial, through some specified degree, in the components of any given irreducible representation of the group SO(3). The results could be applied, for example, to describe the convection patterns in spherically symmetric Rayleigh-Benard convection