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The Dimensions of Irreducible Tensor Representations of the Orthogonal and Symplectic Groups

Published online by Cambridge University Press:  20 November 2018

R. C. King
R. C. King*
Affiliation:
The University, Southampton, England
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It is well known [13] that the irreducible tensor representations (IRs) of the unitary, orthogonal, and symplectic groups in an n-dimensional space may be specified by means of Young tableaux associated with partitions (σ)s = (σ1, σ2, …, σp) with σ1 + σ2 + … + σp = s. Formulae for the dimensions of the corresponding representations have been established [1; 8; 9; 13] in terms of the row lengths of these tableaux. It has been shown [12] for the unitary group, U(n), that the formula may be written as a quotient whose numerator is a polynomial in n containing s factors, and whose denominator is a number independent of n, which likewise may be expressed as a product of s factors. This formula is valid for all n.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 1 , February 1971 , pp. 176 - 188
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Boerner, H., Representations of groups (North Holland, Amsterdam, 1963).Google Scholar
2. Flowers, B. H., The classification of states of the nuclear f-shell, Proc. Roy. Soc. (London) A210 (1952), 497–508.Google Scholar
3. Flowers, B. H., Studies injj coupling. I. Classification of nuclear and atomic states, Proc, Roy. Soc. (London) A212 (1952), 248–263.Google Scholar
4. Flowers, B. H., Studies in jj coupling. IV. The g9/2 shell, Proc. Roy. Soc. (London) A215 (1952), 398–403.Google Scholar
5. Hamermesh, M., Group theory and its application to physical problems (Addison-Wesley, Reading, Massachusetts, 1962).Google Scholar
6. Jahn, H. A., Theoretical studies in nuclear structure. I. Enumeration and classification of the states arising from the filling of the nuclear d-shellt Proc. Roy. Soc. (London) A201 (1950), 516.Google Scholar
7. Judd, B. R., Operator techniques in atomic spectroscopy (McGraw-Hill, New York, 1963).Google Scholar
8. Littlewood, D. E., The theory of group characters, 2nd Ed. (Oxford Univ. Press, Oxford, 1950).Google Scholar
9. Murnaghan, F. D., The theory of group representations (The Johns Hopkins Press, Baltimore, 1938).Google Scholar
10. Newell, M. J.. Modification rules for the orthogonal and symplectic groups, Proc. Roy. Irish Acad. 54 (1951), 153–163.Google Scholar
11. Racah, G., Theory of complex spectra. IV, Phys. Rev. 74 (1949), 1352–1365.Google Scholar
12. Robinson, G. de B., Representation theory of the symmetric group (The University Press, Edinburgh, 1961).Google Scholar
13. Weyl, H., The classical groups (Princeton Univ. Press, Princeton, N.J., 1939).Google Scholar
14. Wybourne, B. G., Group theoretical classification of the atomic states of gN configurations, J. Chem. Phys. 45 (1966), 1100–1104.Google Scholar