Representations of U( N ) are realised as right translations on holomorphic Hilbert (Bargmann) spaces of n x N complex variables. r-fold tensor product spaces of irreducible representations of U ( N ) are shown to be isomorphic to subspaces of the holomorphic Hilbert spaces. Maps are exhibited which carry an irreducible representation of U ( N ) into these subspaces. The algebra of operators commuting with these maps is constructed and it is shown how eigenvalues of certain of these operators can be used to resolve the multiplicity. Several examples from U(3) are explicitly worked out.

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The spectrum of three times ionized scandium, ScIV. R . Smitt Spectrograms from a sliding spark have been taken in a 5 m grazing-incidence spectrograph and a 3 m normal-incidence spectrograph. About 300 lines from 180 8, to 2 500 8, have been identified as ScIV lines and measured with an accuracy of F0.004 8, in the grazing-incidence region (160-320 8,) and of +_O.OOS A in the other region. In addition, about 100 ScIV lines have been identified between 2 350 and 10 000 8, on spectrograms taken by J. E. Holmstrom [5] in this laboratory in a Jarrell-Ash spectrograph having a 21 ft concave grating in Wadsworth mounting. As a result of the analysis almost all the levels belonging to 3s23p54s, 5s, 6s, 7s, 4p, 5p, 3d, 4d, 4J, 5g, 6g, 6h, 7h, 7i and 3s3pa3d are established. By using the polarization formula the ionization limit has been determined from the configurations 5g, 6g, 6h, 7h and 7i to 592 732.413.0 cm-'.