Degeneracy of the SU(3) Direct Product and the Symmetric Representations of SU(6)⊃SU(3)⊗SU(2)

The relation between a state of U(n) associated with an m‐rowed Young diagram, m ≤ n, and a state of U(m) associated with an m‐rowed Young diagram provides the basis for the symmetric state of $\mathrm{U(mn)\supset U(m)\otimes U(n)}$ . As an application, the state vectors associated with the irreducible representations of SU(6) restricted to the subgroup $\text{SU(3)⊗SU(2)}$ are explicitly constructed for the symmetric representation, and generalized to the case of a 2‐rowed Young diagram. Expressions are given for the degeneracy of an $\text{SU(3)⊗SU(2)}$ state in SU(6), and the completeness of the obtained set of states is discussed. The direct product of symmetric $\text{SU(6)⊃SU(3)⊗SU(2)}$ states implies a direct product of SU(3) states; the operator which breaks the degeneracy of the ``2‐rowed'' $\text{SU(6)⊃SU(3)⊗SU(2)}$ state is shown to be Moshinsky's operator X which breaks the degeneracy of the SU(3) direct product.