Bases for Irreducible Representations of the Unitary Group in the Symplectic Group Chain

A method of obtaining the highest weight polynomials of irreducible representations (λ) of Sp_2n occurring in a reduction of an irreducible representation (k) of U_2n is described. The highest weight polynomials of equivalent representations (λ) are labeled by means of parameters which occur naturally from the Littlewood's theorem to determine the branching rules for the representations of the unitary group with respect to the symplectic subgroup, when supplemented by modification rules. The results are given explicitly for $U_4{\mathrm{\supset Sp}}_4$ and $U_6{\mathrm{\supset Sp}}_6,$ the former being a canonical chain and the latter a noncanonical chain.