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We investigate the breaking of SU(3) into its subgroups from the viewpointsof explicit and spontaneous breaking. A one-to-one link between these twoapproaches is given by the complex spherical harmonics, which form a completeset of SU(3)-representation functions. An invariant of degrees p and q incomplex conjugate variables corresponds to a singlet, or vacuum expectationvalue, in a (p,q)-representation of SU(3). We review the formalism of theMolien function, which contains information on primary and secondaryinvariants. Generalizations of the Molien function to the tensor generatingfunctions are discussed. The latter allows all branching rules to be deduced.We have computed all primary and secondary invariants for all proper finitesubgroups of order smaller than 512, for the entire series of groups\Delta(3n^2), \Delta(6n^2), and for all crystallographic groups. Examples ofsufficient conditions for breaking into a subgroup are worked out for theentire T_{n[a]}-, \Delta(3n^2)-, \Delta(6n^2)-series and for allcrystallographic groups \Sigma(X). The corresponding invariants provide analternative definition of these groups. A Mathematica package, SUtree, isprovided which allows the extraction of the invariants, Molien and generatingfunctions, syzygies, VEVs, branching rules, character tables, matrix(p,q)_{SU(3)}-representations, Kronecker products, etc. for the groupsdiscussed above.

Explicit and spontaneous breaking of SU(3) into its finite subgroups