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

Abstract

The basic spin difference character Δ″ 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, associated with partitions κ of k factorize in such a way that Δ″⊗{κ}=(Δ″)r(κ)Πκ, where r(κ) is the Frobenius rank of κ. The analogy between SO(2n) and Sp(2n,R) is shown to be such that the plethysms of the basic harmonic or metaplectic character Δ̃ of Sp(2n,R) factorize in the same way to give Δ̃⊗{κ}=(Δ̃)r(κ)Π̃κ. Moreover, the analogy is shown to extend to the explicit decompositions into characters of irreducible representations of SO(2n) and Sp(2n,R) not only for the plethysms themselves, but also for their factors Πκ and Π̃κ. Explicit 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 Sk. Illustrative 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).