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On the existence of real Clebsch–Gordan coefficients
1.P. H. Butler, Philos. Trans. R. Soc. London, Ser. A 277 (1272), 545 (1975).
2.Reference 1, p. 579.
3.E. P. Wigner, On the Matrices Which Reduce the Kronecker Products of Representations of S. R. Groups. [Unpublished manuscript, 1940;
3.reprinted in Quantum Theory of Angular Momentum, edited by L. C. Biedenharn and H. van Dam (Academic, New York, 1965)].
4.T. Damhus, S. E. Harnung, and C. E. Schäffer (unpublished).
5.W. Feit, Characters of Finite Groups (Benjamin, New York, 1967), pp. 38–40.
6.L. Nachbin, The Haar Integral (van Nostrand, New York, 1965), p. 81, Proposition 13.
7.E. P. Wigner, SIAM J. Appl. Math. 25, 169 (1973).
8.A. A. Kirillov, Elements of The Theory of Representations (Springer‐Verlag, Berlin, 1976), p. 133, Theorem 1.
9.A. A. Kirillov, Ref. 8, p. 139, Theorem 1.
10.L. Nachbin, Ref. 6, p. 83, Proposition 16 (put to obtain
11.E. P. Wigner, Group Theory and its Application to the Quantum Mechanics of Atomic Spectra (Academic, New York, 1959), pp. 78–79. [The proof given here is unnecessarily complicated. Given a nonsingular matrix M intertwining (connecting) two unitary irreducible matrix representations of any group, the product of Mand its adjoint matrix can be shown from Schur’s lemma to be a scalar matrix. Since this product matrix obviously can have only positive real eigenvalues we conclude that it is a positive real multiple of the unit matrix, so that M itself is proportional to a unitary matrix].
12.F. G. Frobenius and I. Schur, Sitzungsber. Königl. Preuss. Akad. Wiss. (Berlin) 1906, 186.
13.E. P. Wigner, Ref. 11, pp. 285–289.
14.L. Jansen and M. Boon, Theory of Finite Groups. Applications in Physics (North‐Holland, Amsterdam, 1967), pp. 125–132.
15.I. M. Isaacs, Character Theory of Finite Groups (Academic, New York, 1976), Chap. 4.
16.W. Feit, Ref. 5, pp. 20–23;
16.A. A. Kirillov, Ref. 8, Secs. 8.2 and 11.1.
17.G. W. Mackey, Amer. J. Math. 75, 387 (1953).
18.T. Damhus, Linear Algebra Appl. (to be published).
19.W. Feit, Ref. 5, p. 23;
19.E. P. Wigner, Ref. 7, p. 181;
19.I. M. Isaacs, Ref. 15, pp. 49–50.
20.E. P. Wigner, Amer. J. Math. 63, 57 (1941).
21.U. Fano and G. Racah, Irreducible Tensorial Sets (Academic, New York, 1959), p. 18 and pp. 23–24.
22.S. E. Harnung, Molecular Phys. 26, 473 (1973).
23.U. Fano and G. Racah, Ref. 21, p. 18 and p. 33.
24.P. H. Butler and B. G. Wybourne, Internat. J. Quant. Chem. 10, 581 (1976);
24.P. H. Butler, in Recent Advances in Group Theory and Their Application to Spectroscopy, edited by J. C. Donini (Plenum, New York, 1979), p. 155.
25.P. H. Butler and R. C. King, Can. J. Math. 26, 328 (1974).
26.J. S. Frame (personal communication, August 1978).
27.W. Feit, Ref. 5, p. 17;
27.I. M. Isaacs, Ref. 15, p. 16, Theorem 2.8.
28.W. T. Sharp, L. C. Biedenharn, E. de Vries, and A. J. van Zanten, Can. J. Math. 27, 246 (1975).
29.I. M. Isaacs, Ref. 15, pp. 57–58;
29.U. Fano and G. Racah, Ref. 21, Appendix C, p. 134.
30.T. Damhus and S. E. Harnung (unpublished).
31.P. Landrock (personal communication, August 1979).
32.See, e.g., L. Jansen and M. Boon, Ref. 14, pp. 46–50.
33.Specifically, the quaternion group is embedded in the automorphism group of by assigning to any two fourth‐order elements which are not mutually inverse the GL(2, 3)‐matnces and respectively, and letting GL(2, 3)‐matrices operate on in the natural way ( is viewed as a vector space over the finite field of order 3). By this construction the quaternion group acts transitively on
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