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Calculation of the Inner Multiplicity of Weights by Means of the Branching‐Law Method and by Racah's Recurrence Relation

### Abstract

We discuss here in detail the validity of the branching‐law method suggested in a previous paper [M. K. F. Wong, J. Math. Phys. **11,** 1489 (1970)] for the calculation of the inner multiplicity of weights in an irreducible representation of a classical group. It is found that the method works for all the irreducible representations of *SU*(*n*) and *SO*(2*n* + 1), but that in the case of *SO*(2*n*) and *Sp*(2*n*) the method does not always give complete solutions except in some simple cases. It is then suggested that Racah's recurrence relation be used in these cases so that complete solutions may be obtained. It is also noted that Racah's recurrence relation alone is sufficient to obtain the inner multiplicity of all weights. This fact is utilized in the calculation of inner multiplicities in another paper [B. Gruber, J. Math. Phys. **11,** 3077 (1970)]. The method suggested in this paper is illustrated through the calculation of some typical examples of the inner multiplicity of weights in the two classical groups*SO*(2*n*) and *Sp*(2*n*).

© 1970 The American Institute of Physics

Received 09 March 1970
Published online 28 October 2003