Orbit spaces of low-dimensional representations of simple compact connected Lie groups and extrema of a group-invariant scalar potential

Journal Article · Fri Jun 01 00:00:00 EDT 1984 · J. Math. Phys. (N.Y.); (United States)

DOI:https://doi.org/10.1063/1.526347· OSTI ID:6760363

Kim, J S

Orbit spaces of low-dimensional representations of classical and exceptional Lie groups are constructed and tabulated. We observe that the orbit spaces of some single irreducible representations (adjoints, second-rank symmetric and antisymmetric tensors of classical Lie groups, and the defining representations of F/sub 4/ and E/sub 6/) are warped polyhedrons with (locally) more protrudent boundaries corresponding to higher level little groups. The orbit spaces of two irreducible representations have different shapes. We observe that dimension and concavity of different strata are not sharply distinguished. We explain that the observed orbit space structure implies that a physical system tends to retain as much symmetry as possible in a symmetry breaking process. In Appendix A, we interpret our method of minimization in the orbit space in terms of conventional language and show how to find all the extrema (in the representation space) of a general group-invariant scalar potential monotonic in the orbit space. We also present the criterion to tell whether an extremum is a local minimum or maximum or an inflection point. In Appendix B, we show that the minimization problem can always be reduced to a two-dimensional one in the case of the most general Higgs potential for a single irreducible representation and to a three-dimensional one in the case of an even degree Higgs potential for two irreducible representations. We explain that the absolute minimum condition prompts the boundary conditions enough to determine the representation vector.

• Research Organization: California Institute of Technology (CalTech), Pasadena, CA (United States)
• DOE Contract Number: AC03-81ER40050
• OSTI ID: 6760363
• Journal Information: J. Math. Phys. (N.Y.); (United States), Vol. 25:6
• Country of Publication: United States
• Language: English

Related Subjects

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
FIELD THEORIES
SCALAR FIELDS
SYMMETRY BREAKING
LIE GROUPS
IRREDUCIBLE REPRESENTATIONS
GAUGE INVARIANCE
HIGGS MODEL
RENORMALIZATION
THEORETICAL DATA
DATA
INFORMATION
INVARIANCE PRINCIPLES
MATHEMATICAL MODELS
NUMERICAL DATA
PARTICLE MODELS
SYMMETRY GROUPS
645400* - High Energy Physics- Field Theory