1. G. Frobenius and I. Schur, Sitzber. Deut. Akad. Wiss. Berlin, Kl. Math. Phys. Tech., p. 186 (1906).
2. Cf. F. D. Murnaghan, “Group Representations,” Chapter 3 ( Baltimore, Maryland, 1938 ).
3. The parametrization of the group and the form (29), (30a) of the representations is identical with that adopted in E. Wigner, “Gruppentheorie und ihre Anwendungen auf etc.” ( Braunschweig, 1931 ), Chapter XV. This chapter also gives the isomorphism of the three-dimensional rotation and two-dimensional unimodular unitary groups.
4. j’ + ½) + (j’ − ½) denotes the sum of the representations j’ + ½ and j’ − ½ (i.e. the reducible representation containing j’ + ½ and j’ − ½ . Cf. F. D. Murnaghan, Reference 2.
5. A collection of formulas for these quantities (which are essentially identical with the three-j-symbols) is given in “The Theory of Atomic Spectra” by E. U. Condon and G. H. Shortley (Cambridge Univ. Press, London and New York, 1935), Chapter III. Relations of this character are, of course, well known in quantum mechanics and many of the six -j-symbols are implicitly contained in the formulas for the relative intensities of spectral lines and the magnitude of the Zeeman effect (Hönl, Kronig, Rubinowicz). Cf. the monographs of Reference 3 and 5, particularly the latter.