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

TL;DR

This paper explores how SU(3) symmetry can be explicitly or spontaneously broken into its finite subgroups using invariants, providing computational tools and a comprehensive analysis of subgroup invariants and breaking conditions.

Contribution

It introduces a complete formalism linking explicit and spontaneous breaking of SU(3), computes invariants for subgroups, and provides a Mathematica package for group analysis.

Findings

Computed all invariants for subgroups under 512 elements.

Established conditions for symmetry breaking into various subgroups.

Provided tools for analyzing group invariants and representations.

Abstract

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 allows all branching rules to be deduced. We have computed all primary and secondary invariants for all proper finite subgroups of order smaller than 512, for the entire series of groups \Delta(3n^2), \Delta(6n^2), and for all crystallographic groups. Examples of sufficient conditions for breaking into a subgroup are worked out for the entire T_{n[a]}-, \Delta(3n^2)-, \Delta(6n^2)-series and for all crystallographic groups \Sigma(X). The corresponding invariants provide an alternative definition of these groups. 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.