Welcome to the new version of CaltechAUTHORS. Login is currently restricted to library staff. If you notice any issues, please email coda@library.caltech.edu
Published June 1, 1984 | public
Journal Article Open

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

Kim, Jai Sam

Abstract

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 F4 and E6) 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.

Additional Information

Copyright © 1984 American Institute of Physics. (Received 7 September 1983; accepted 4 November 1983) I would like to thank Professor S. Frautschi for carefully reading the manuscripts. I am grateful to Dr. G. Sartori and Dr. M. Jarić for sending me their papers and illustrating their methods. I would like to thank Professor F. Gürsey for illuminating discussions and extensive references and for introducing me to mathematicians, Professor G. Zuckerman of Yale University and Professor Jacob Towber of De Paul University, who gave man an excellent lecture on classical invariant theory. I would like to thank Professor L. Michel for informing me of counterexamples to the Michel-Radicati conjecture and for encouraging me. I would like to thank Professor K.S. Kang for constructive criticism. I would like to thank Professor G. Schwarz of Brandeis University for explaining his works and Professor W.-C. Hsiang of Princeton University for helpful discussions and relevant reprints. I would like to thank Dr. M. Roček and the theoretical physics group at Stony Brook and Professor C.M> Bender for encouragement.

Files

KIMjmp84.pdf
Files (1.8 MB)
Name Size Download all
md5:301572cdf1e47de67b6b8579f8438ee7
1.8 MB Preview Download

Additional details

Created:
August 22, 2023
Modified:
October 16, 2023