PRODUCTS AND SYMMETRIZED POWERS OF IRREDUCIBLE REPRESENTATIONS OF SP(2N, R) AND THEIR ASSOCIATES

The calculation of Kronecker products and plethysms of the infinite-di mensional harmonic series unitary irreducible representations of the n on-compact group Sp(2n, K) is considered. The complementarity of Sp(2n , K) and O(k) is used to define associate irreducible representations of Sp(2n, K). This leads to simple relationships between Kronecker pro ducts and plethysms of irreducible representations of Sp(2n, K) and th ose of their corresponding associate irreducible representations. In t he process of proving the validity of these previously conjectured rel ationships several new identities are found for plethysms involving in finite series of Schur functions. In addition, a general formula for p lethysms of arbitrary irreducible representations of Sp(2n, K is deriv ed and its implementation is illustrated with a detailed example. A re markable analogy is then observed between plethysms of the basic harmo nic irreducible representations of Sp(2n, K) and those of the basic sp in irreducible representations of SO(2n).