Analogies between finite-dimensional irreducible representations of SO(2n)and infinite-dimensional irreducible representations of Sp(2n,R). II. Plethysms

The basic spin difference character Delta" of SO(2n) is a useful device in dealing with characters of irreducible spinor representations of SO(2n). It is shown here that its kth-fold symmetrized powers, or plethysms, associat ed with partitions kappa of k factorize in such a way that Delta"x{kappa}=( Delta")(r(kappa))Pi(kappa), where r(kappa) is the Frobenius rank of kappa. The analogy between SO(2n) and Sp(2n,R) is shown to be such that the plethy sms of the basic harmonic or metaplectic character <(Delta)over tilde> of S p(2n,R) factorize in the same way to give <(Delta)over tilde>x{kappa}=(<(De lta)over tilde>)(r(kappa))<(Pi)over tilde>(kappa). Moreover, the analogy is shown to extend to the explicit decompositions into characters of irreduci ble representations of SO(2n) and Sp(2n,R) not only for the plethysms thems elves, but also for their factors Pi(kappa) and <(Pi)over tilde>(kappa). Ex plicit formulas are derived for each of these decompositions, expressed in terms of various group-subgroup branching rule multiplicities, particularly those defined by the restriction from O(k) to the symmetric group S-k. Ill ustrative examples are included, as well as an extension to the symmetrized powers of certain basic tensor difference characters of both SO(2n) and Sp (2n,R). (C) 2000 American Institute of Physics. [S0022-2488(00)02608-6].