Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-12-02T23:55:47.795Z Has data issue: false hasContentIssue false
  • English
  • Français

On Quasi-Ambivalent Groups

Published online by Cambridge University Press:  20 November 2018

W. T. Sharp ,
L. C. Biedenharn ,
E. De Vries  and
A. J. Van Zanten
W. T. Sharp
Affiliation:
University of Toronto, Toronto, Ontario
L. C. Biedenharn
Affiliation:
University of Toronto, Toronto, Ontario
E. De Vries
Affiliation:
Duke University, Durham, North Carolina
A. J. Van Zanten
Affiliation:
Duke University, Durham, North Carolina
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The prototype for applications of group theory to physics, and to mathematical physics, is the quantum theory of angular momentum [1] ; the use of such techniques is now almost universal, and familiarly (through somewhat imprecisely) known as “Racah algebra”. To categorize, group theoretically, those characteristics which underlay this applicability to physical problems, Wigner [30] isolated two significant conditions, and designated groups possessing these properties as simply reducible.

The two conditions for simple reducibility are:

(a)Every element is equivalent to its reciprocal, i.e., all classes are ambivalent.

(b) The Kronecker (or “direct“) product of any two irreducible representations of the group contains no representation more than once.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 2 , 01 April 1975 , pp. 246 - 255
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Biedenharn, L. C. and van Dam, H. (eds.), Quantum theory oj angular momentum (Academic Press, New York-London, 1965).Google Scholar
2. Biedenharn, L. C., Nuyts, J., and Ruegg, H., On generalizations of isoparity, Comm. Math. Phys. 2 (1966), 231–250.Google Scholar
3. Biedenharn, L. C., Giovannini, A., and Louck, J. D., Canonical definition of Wigner coefficients in Uni J. Mathematical Phys. 8 (1967), 691–700.Google Scholar
4. Biedenharn, L. C., Brouwer, W., and Sharp, W. T., The algebra of representations of some finite groups, Rice University Studies 54, no. 2 (1968).Google Scholar
5. Biedenharn, L. C., Louck, J. D., Chacon, E., and Ciftan, M., On the structure of the canonical tensor operators in the unitary groups. I, An extension of the pattern calculus rules and the canonical splitting in £7(3), J. Mathematical Phys. 13 (1972), 1957–1984.Google Scholar
6. Biedenharn, L. C. and Louck, J. D., On the structure of the canonical tensor operators in the unitary groups. II, The tensor operators in £7(3) characterized by maximum null space, J. Mathematical Phys. 13 (1972), 1985–2001.Google Scholar
7. Boerner, H., Representations of groups, 2nd ed. (North-Holland Publ. Comp., Amsterdam- London, 1970).Google Scholar
8. Bose, A. K. and Patera, J., Classification of finite-dimensional irreducible representations of connected complex semisimple Lie groups, J. Mathematical Phys. 7 (1970), 2231–2234.Google Scholar
9. Burnside, W., Theory of groups of finite order, 2nd ed. (Dover Publ., New York, 1955).Google Scholar
10. Butler, P. H. and King, R. C., Symmetrized kronecker products of group representations, Can. J. Math. 26 (1974), 328–339.Google Scholar
11. Coxeter, H. S. M. and Moser, W. O. J., Generators and relations for Discrete groups, 2nd ed. (Springer Verlag, Berlin-Gôttingen-Heidelberg-New York, 1965).Google Scholar
12. Derome, J.-R. and Sharp, W. T., Racah algebra for an arbitrary group, J. Mathematical Phys. 6 (1965), 1584–1590.Google Scholar
13. Derome, J.-R., Symmetry properties of the 3;-symbols for an arbitrary group, J. Mathematical Phys. 7 (1966), 612–615.Google Scholar
14. Derome, J.-R., Symmetry properties of the 3j-symbols for SU (3), J. Mathematical Phys. 8 (1967), 714–716.Google Scholar
15. Feit, W., Characters of finite groups (W. A. Benjamin, New York-Amsterdam, 1967).Google Scholar
16. Frobenius, G. and Schur, I., Uber die reellen Darstellingen der endlichen Gruppen, Sitzungsberichte der kôn (preuss. Ak. der Wissenschaften, Jahrgang 1906, S. 186, Berlin).Google Scholar
17. Gorenstein, D., Finite groups (Harper and Row Publ., New York-Evanston-London, 1968).Google Scholar
18. Hamermesh, M., Group theory and its applications to physical problems (Addison-Wesley Publ. Comp., Reading (Mass.), 1964).Google Scholar
19. Kurosh, A. G., The theory of groups, 2nd ed. vol. 1 (Chelsea Publ. Comp., New York, 1960).Google Scholar
20. Louck, J. D., Recent progress toward a theory of tensor operators in the unitary groups, Amer. J. Phys. 38 (1970), 3–42.Google Scholar
21. Mackey, G. W., Multiplicity free representations of finite groups, Pacific J. Math. 8 (1958), 503–510.Google Scholar
22. Mal'cev, A. I., On semi-simple subgroups of Lie groups, Izv. Akad. Nauk SSSR Ser. Mat. 8 (1944), 143–174 [in Amer. Math. Soc. Transi, no. 33 (1950)].Google Scholar
23. Mehta, M. L., Classification of irreducible unitary representations of compact simple Lie groups. I, J. Mathematical Phys. 7 (1966), 1824–1832.Google Scholar
24. Mehta, M. L. and Srivastava, P. K., Classification of irreducible unitary representations of compact simple Lie groups. II, J. Mathematical Phys. 7 (1966), 1833–1835.Google Scholar
25. Murnaghan, F. D., The theory of group representations (Dover Publ., New York, 1963).Google Scholar
26. Sharp, W. T., Racah algebra and the contraction of groups (AECL-1098, Atomic Energy of Canada Limited, Chalk River, Ont., 1960; a revised version, edited by E. de Vries and A. J. van Zanten is to be published by University of Toronto Press).Google Scholar
27. Speiser, A., Die Théorie der Gruppen von endlicher Ordnung 4te Aufiage (Birkhàuser Verlag, Basel-Stuttgart, 1956).Google Scholar
28. van Zanten, A. J., Some applications of the representation theory of finite groups: a partial reduction method, Ph.D. thesis, Groningen, 1972.Google Scholar
29. van Zanten, A. J. and de Vries, E., On the number of r∞ts of the equation Xn = lin finite groups and related properties, J. Algebra 25 (1973), 475–486.Google Scholar
30. Wigner, E. P., On representations of certain finite groups, Amer. J. Math. 63 (1941), 57–63 (reprinted in [1]).Google Scholar
31. Wigner, E. P., On the matrices which reduce the Kronecker products of representations of S. R. groups (Princeton, 1951, reprinted in [1]).Google Scholar
32. Wigner, E. P., Group theory and its application to the quantum mechanics of atomic spectra (Academic Press, New York-London, 1959).Google Scholar