Decomposition of Tensors of the Classical Groups

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Abstract
A tensor symmetrization procedure obtained in a recent publication [Phys. Rev. Letters 16, 1058 (1966)] is shown to support rather than disprove Weyl's tensor symmetrization theorem. This ``extended'' symmetrization procedure differs from Weyl's approach in that to construct a subspace irreducible under GL(n, c) one starts with a set of formal states (symmetrized tensors with formal index values) spanning an irreducible representation of the permutation group rather than starting with a single formal state. Extended symmetrization is often more useful than Weyl's approach because the states obtained are highly organized and because it also yields an efficient independent state selection method for the symmetrization procedures using modified Young symmetrizers and Wigner projection operators. The state organization obtained makes it possible to show that the nonorthogonality which is present for bases obtained with Young symmetrizers can be easily removed. The state organization also makes it possible to simplify the task of recoupling symmetrized tensor representations to gain a simply‐coupled form. This form enlarges the class of Clebsch‐Gordan and recoupling coefficients which can be evaluated by tensor methods. Group matrices and Lie group generator matrix elements are also obtained by tensor methods. Extended symmetrization using unitary representation Wigner projection operators based on unitary representations is shown to result in orthogonal states although usually not the orthogonal states desired. The usual Young symmetrizers are shown to often be more useful than modified Young symmetrizers or Wigner projection operators.

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