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

Preprint number: CP3-Origins-2011-33 and DIAS-2011-25
Authors: Alexander Merle (University of Stockholm) and Roman Zwicky (CP3-Origins)
External link: arXiv.org

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We investigate the breaking of SU(3) into its subgroups from the viewpoints of explicit and spontaneous breaking. A one-to-one link between these two approaches is given by the complex spherical harmonics, which form a complete set of SU(3)-representation functions. An invariant of degrees p and q in complex conjugate variables corresponds to a singlet, or vacuum expectation value, in a (p,q)-representation of SU(3). We review the formalism of the Molien function, which contains information on primary and secondary invariants. Generalizations of the Molien function to the tensor generating functions are discussed. The latter allow to deduce all branching rules. We have computed all primary and secondary invariants for all proper finite subgroups of order smaller than 512, for the entire series of groups ∆(3n2), ∆(6n2), and for all crystallgraphic groups. Examples of sufficient conditions for breaking into a subgroup are worked out for the entire Tn[a]-, ∆(3n2)-, ∆(6n2)-series and for all crystallographic groups Σ(X). A Mathematica package, SUtree, is provided which allows the extraction of the invariants, Molien and generating functions, syzygies, VEVs, branching rules, character tables, matrix (p,q)SU(3)-representations, Kronecker products, etc. for the groups discussed above.