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

Kim, Jai Sam (1984) Orbit spaces of low-dimensional representations of simple compact connected Lie groups and extrema of a group-invariant scalar potential. Journal of Mathematical Physics, 25 (6). pp. 1694-1717. ISSN 0022-2488. doi:10.1063/1.526347.

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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.

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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.
Subject Keywords:Lie groups; irreducible representations; field theories; gauge invariance; symmetry breaking; renormalization; scalar fields; higgs model; theoretical data
Issue or Number:6
Record Number:CaltechAUTHORS:KIMjmp84
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9111
Deposited By: Archive Administrator
Deposited On:22 Apr 2008
Last Modified:08 Nov 2021 20:56

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