Symmetrized Kronecker Products of Group Representations

Butler, P. H. ; King, R. C.

Canadian journal of mathematics, Tome 26 (1974) no. 2, pp. 328-339

Résumé

Certain phases are associated with the Kronecker squares and cubes of representations of the finite and of the compact semi-simple groups. These phases are important in giving the symmetry properties of the 1 — jm and 3 — jm symbols of the groups [4; 9]. It is our primary purpose to evaluate these phases.The Frobenius-Schur invariant [12, p. 142] for an irreducible representation of group G (1.1)

Détail

DOI : 10.4153/CJM-1974-034-x

Bibliographie

[1] 1. Abramsky, J. and King, R. C., Formation and decay of negative-parity baryon resonances in a broken U model, Nuovo Cimento A 67 (1970), 153–216. Google Scholar

[2] 2. Biedenharn, L. C., Brouwer, W., and Sharp, W. T., The algebra of representations of some finite groups (Rice Univ. Studies, Vol. 54, No. 2, 1968). Google Scholar

[3] 3. Bose, A. K. and Patera, J., Classification of finite-dimensional irreducible representations of connected complex semi-simple Lie groups, J. Mathematical Phys. 11 (1970), 2231–2234. Google Scholar

[4] 4. Butler, P. H., Wigner coefficients and n — j symbols for chains of groups (to appear). Google Scholar

[5] 5. Butler, P. H. and King, R. C., The symmetric group: characters, products and plethysms, J. Mathematical Phys. 14 (1973), 1176–1183. Google Scholar

[6] 6. Derome, J-R., Symmetry properties of the 3j-symbols for an arbitrary group, J. Mathematical Phys. 7 (1966), 612–615. Google Scholar

[7] 7. Derome, J-R., Symmetry properties of Sj-symbols for SU ‘(3), J. Mathematical Phys. 8 (1967), 714–716. Google Scholar

[8] 8. Derome, J-R. and Jabimow, G., Propriétés de symétrie des symboles à 3j de SU(n), Can. J. Phys. 48 (1970), 2169–2175. Google Scholar

[9] 9. Derome, J-R. and Sharp, W. T., Racah algebra for an arbitrary group, J. Mathematical Phys. 6 (1965), 1584–1590. Google Scholar

[10] 10. Dynkin, E. B., The maximal subgroups of the classical groups, Amer. Math. Soc. Transi. Ser. 26 (1957), 245–378. Google Scholar

[11] 11. Frame, J. S., private communication, June 1972. Google Scholar

[12] 12. Hamermesh, M., Group theory and its application to physical problems (Addison-Wesley Pub. Co., Reading, Mass., 1962). Google Scholar

[13] 13. King, R. C., Generalised Young tableaux and the general linear group, J. Mathematical Phys. 11 (1970), 280–294. Google Scholar

[14] 14. King, R. C., Modification rules and products of irreducible representations of the unitary, orthogonal and symplectic groups, J. Mathematical Phys. 12 (1971), 1588–1598. Google Scholar

[15] 15. King, R. C., Branching rules for GL(N) ⊃ Σ and the evaluation of inner plethysms, J. Mathematical Phys. (to appear). Google Scholar

[16] 16. Littlewood, D. E., Modular representations of symmetric groups, Proc. Roy. Soc. London Ser. A209 (1951), 333–352. Google Scholar

[17] 17. Littlewood, D. E., Products and plethysms of characters with orthogonal, symplectic and symmetric groups, Can. J. Math. 10 (1958), 17–32. Google Scholar

[18] 18. Littlewood, D. E., The theory of group characters, 2nd edition (Oxford University Press, Oxford, 1950). Google Scholar

[19] 19. Mal'cev, A. I., On semi-simple subgroups of Lie groups, Amer. Math. Soc. Transi. Ser. 1 9 (1962), 172–213. Google Scholar

[20] 20. Plunkett, S. P. O., On the plethysm of S-functions, Can. J. Math. 24 (1972), 541–552. Google Scholar

[21] 21. Robinson, G. de B., Representation theory of the symmetric group (University of Toronto, Toronto, 1961). Google Scholar

[22] 22. Wybourne, B. G., Symmetry principles and atomic spectroscopy (with an appendix of tables by P. H. Butler) (J. Wiley and Sons, New York, 1970). Google Scholar

[23] 23. van Zanten, A. J. and de Vries, E., On the number of roots of the equation X = E in finite groups and related properties, J. Algebra 25 (1973), 475–486. Google Scholar