Unitary Groups: Representations and Decompositions

C. ITZYKSON; M. NAUENBERG

doi:10.1103/RevModPhys.38.95

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Unitary Groups: Representations and Decompositions

C. ITZYKSON^* and M. NAUENBERG^†

C. ITZYKSON^*

Stanford Linear Accelerator Center, Stanford University, Stanford, California

M. NAUENBERG^†

Department of Physics, Stanford University, Stanford, California

^*On leave from Service de Physique Theorique, CEN Saclay, BP No. 2, Gif sur Yvette (S. et O.), France.
^†A. P. Sloan Fellow.

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Rev. Mod. Phys. 38, 95 – Published 1 January, 1966

DOI: https://doi.org/10.1103/RevModPhys.38.95

Abstract

An elementary account is given of the representation theory for unitary groups. We review the basic definitions and the construction of irreducible representations using tensor methods, and indicate the connection to the infinitesimal approach. Special attention has been given to the detailed procedure to obtain Clebsch-Gordan series and to the problem of finding the ( $S U_{m}, S U_{n}$ ) content of an irreducible representation of $S U_{mn}$ or $S U_{m + n}$ . An appendix summarizes the properties of the Young operators used in constructing the tensor representations; this provides the link with the representation theory of the symmetric groups. We include a tabulation of various decompositions which appear in the text and of Weyl's dimension formula for tensor representation.

References (19)

H. Weyl, Classical Groups (Princeton University Press, Princeton, New Jersey, 1946) H. WeylThe Theory of Groups and Quantum Mechanics (Dover Publications, Inc., New York, 1950)
F. D. Murnaghan, The Theory of Group Representations (Dover Publications, Inc., New York, 1963)
D. Littlewood, Theory of Group Characters and Matrix Representations of Groups (Oxford University Press, New York, 1950)
M. Hamermesh, Group Theory and Its Application to Physical Processes (Addison-Wesley Publishing Company, Reading, Massachusetts, 1962)
G. Racah, Group Theory and Spectroscopy (Princeton University Press, Princeton, New Jersey, 1951)
M. L. Whippman, "Branching Rules for Simple Lie Groups," University of Pennsylvania preprint (1964)
C. R. Hagen and A. J. MacFarlane, "Reduction of Representations of $S U_{mn}$ and $S U_{m + n}$ with Respect to the Subgroup $S U_{m}$ , $S U_{n}$ ," Syracuse University preprint (1965)
Omitted endnote
Ref. 1, Chaps. III and IV
Omitted endnote
Omitted endnote
Ref. 1, p. 201
Omitted endnote
Omitted endnote
Omitted endnote
Omitted endnote
Omitted endnote
Omitted endnote
Ref. 4

[c1] H. Weyl, Classical Groups (Princeton University Press, Princeton, New Jersey, 1946) H. WeylThe Theory of Groups and Quantum Mechanics (Dover Publications, Inc., New York, 1950)

[c2] F. D. Murnaghan, The Theory of Group Representations (Dover Publications, Inc., New York, 1963)

[c3] D. Littlewood, Theory of Group Characters and Matrix Representations of Groups (Oxford University Press, New York, 1950)

[c4] M. Hamermesh, Group Theory and Its Application to Physical Processes (Addison-Wesley Publishing Company, Reading, Massachusetts, 1962)

[c5] G. Racah, Group Theory and Spectroscopy (Princeton University Press, Princeton, New Jersey, 1951)

[c6] M. L. Whippman, "Branching Rules for Simple Lie Groups," University of Pennsylvania preprint (1964)

[c7] C. R. Hagen and A. J. MacFarlane, "Reduction of Representations of $S U_{mn}$ and $S U_{m + n}$ with Respect to the Subgroup $S U_{m}$ , $S U_{n}$ ," Syracuse University preprint (1965)

[c8] Omitted endnote

[c9] Ref. 1, Chaps. III and IV

[c10] Omitted endnote

[c11] Omitted endnote

[c12] Ref. 1, p. 201

[c13] Omitted endnote

[c14] Omitted endnote

[c15] Omitted endnote

[c16] Omitted endnote

[c17] Omitted endnote

[c18] Omitted endnote

[c19] Ref. 4

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Unitary Groups: Representations and Decompositions

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References (19)

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