Symmetry breaking patterns for third-rank totally antisymmetric representations of unitary groups

Some possible symmetry breaking patterns for unitary group gauge theoretic models based on Higgs scalars in the third-rank totally antisymmetric tensor representations of U(n) are studied. The critical points are expressed in terms of a single parameter xi such that 1/n<or= xi <or=1/3. It is shown that spontaneous symmetry breaking may take place from SU(n) to SU(3)*SU(n-3) for all n, from SU(6) to SU(3)*SU(3), from SU(7) to G₂ from SU(8) to SU(3) and from SU(9) to U(1)*U(1)*U(1)*U(1). The absolute minima are shown to be degenerate for n>or=9, and remarks are made concerning a conjecture of Michel (1979).