Standard Young tableaux and weight multiplicities of the classical Lie groups

By examining the branching rules for all irreducible representations of the classical groups U(k), SU(k), SO(2k+1), Sp(2k) and SO(2k) on restriction to U(1)*U(1)*U(1), standard Young tableaux are specified for each of these groups. It is shown that these tableaux determine the corresponding characters of the irreducible representations. The rules for constructing these tableaux are derived and in this way the determination of weight multiplicities is reduced to a simple combinatorial exercise. General formula for such weight multiplicities are given encompassing the most difficult case: namely that of SO(2k). Illustrative examples are provided, including some yielding the explicit k-dependence of weight multiplicities.

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