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Criteria for Groups with Representations of the Second Kind and for Simple Phase Groups

  • A. J. Van Zanten (a1) and E. De Vries (a1)

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In this paper we consider matrix representations of compact groups over the field of the complex numbers. We shall deal mainly with finite groups.

The Kronecker product of two irreducible representations σ1 and σ2 of a group is in general a reducible representation of . The explicit reduction of such a product to irreducible representations σ3 can be performed by means of a unitary matrix, the elements of which are called Wigner coefficients or Clebsch-Gordan coefficients [1; 25; 27].

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References

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1. Biedenharn, L. C. and van Dam, H. (eds.), Quantum theory of angular momentum (Academic Press, New York-London, 1965).
2. Biedenharn, L. C., Nuyts, J., and Ruegg, H., On generalizations of isoparity, Comm. Math. Phys. 2 (1966), 231250.
3. Bose, A. K. and Patera, J., Classification of finite-dimensional irreducible representations of connected complex semisimple Lie groups, J. Mathematical Phys. 7 (1970), 22312234.
4. Burnside, W., Theory of groups of finite order, 2nd edition (Dover Publ., New York, 1955),
5. Burrow, M., Representation theory of finite groups (Academic Press, New York-London, 1965).
6. Butler, P. H., Wigner coefficients and n — j symbols for chains of groups (to appear).
7. Butler, P. H. and King, R. C., Symmetrized Kronecker products of group representations, J. Mathematical Phys. U (1973), 11761183.
8. Coxeter, H. L.M. and Moser, W. O. J., Generators and relations for discrete groups, 2nd edition (Springer Verlag, Berlin-Gottingen-Heidelberg-New York, 1965).
9. Derome, J.-R. and Sharp, W. T., Racah algebra for an arbitrary group, J. Mathematical Phys. 6 (1965), 15841590.
10. Derome, J.-R., Symmetry properties of the Sj-symbols for an arbitrary group, J. Mathematical Phys. 7 (1966), 612615.
11. Feit, W., Characters of finite groups (W. A. Benjamin, New York-Amsterdam, 1967).
12. Frobenius, G. and Schur, I., Uber die reellen Darstellungen der endlichen Gruppen, Sitzungsberichte der kôn. preuss. Ak. der Wissenschaften (1906), 186208.
13. Hall, M. Jr., The theory of groups (Macmillan Co., New York, 1964).
14. Hall, M. Jr., and Wales, D., The simple group of order 604,800, J. Algebra 9 (1968), 417450.
15. Huppert, B., Endliche Gruppen, I (Springer Verlag, Berlin-Heidelberg-New York, 1955).
16. King, R. C., Branching rules far (GL(N) 2) £ m and the evaluation of inner plethysms, J. Mathematical Phys. 15 (1974), 258267.
17. Ledermann, W., Introduction to the theory of finite groups, 5th edition (Oliver and Boyd, Edinburgh-London, 1964).
18. Mackey, G. W., Symmetric and antisymmetric Kronecker squares and intertwining numbers of induced representations of finite groups, Amer. J. Math. 75 (1953), 387405.
19. Mal'cev, A. I., On semi-simple subgroups of Lie groups, Izv. Akad. Nauk. SSSR ser. Mat. 8 (1944), 143174 [Am. Math. Soc. Transi, no. 23 (1950)].
20. Mehta, M. L., Classification of irreducible unitary representations of compact simple Lie groups. I, J. Mathematical Phys. 7 (1966), 18241832.
21. Mehta, M. L. and Srivastava, P. K., Classification of irreducible representations of compact simple Lie groups. II, J. Mathematical Phys. 7 (1966), 18331835.
22. Schur, I., Arithmetische Untersuchungen uber endliche Gruppen linearer Substitutionen, Sitzungsberichte der kon. preuss. Ak. der Wissenschaften (1906), 164184 (Berlin).
23. van Zanten, A. J., Some applications of the representation theory of finite groups: a partial reduction method, Ph.D. thesis, Groningen, 1972.
24. van, A. J. Zanten and de Vries, E., On the number of roots of the equation Xn= 1 infinite groups and related properties, J. Algebra 25 (1973), 475486.
25. van, A. J. Zanten and de Vries, E., On the number of classes of a finite group invariant for certain substitutions, Can. J. Math. 26 (1974), 10901097.
26. Wigner, E. P., On the matrices which reduce the Kronecker products of representations of S.R. groups (Princeton, 1951, reprinted in ref. 1).
27. Wigner, E. P., Group theory and its application to the quantum mechanics of atomic spectra (Academic Press, New York-London, 1959).
28. Wigner, E. P., On representations of certain finite groups, Amer. J. Math. 63 (1941), 5763 (reprinted in ref. 1).
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Criteria for Groups with Representations of the Second Kind and for Simple Phase Groups

  • A. J. Van Zanten (a1) and E. De Vries (a1)

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