The Gel’fand states of certain representations of U (n) and the decomposition of products of representations of U (2)

Abstract

The representations of U (n), as realized by Bargmann and Moshinsky on spaces of polynomials (’’boson calculus’’), are the main subject of this paper. We consider them from a global point of view, pointing out the connection with induced representations. To compute the detailed structure of the representations, we find the reproducing kernels of the function spaces and the operators that connect them according to Weyl’s branching law. Using these results, we compute the boson polynomials of representations of U (3), and arrange them in a generating function. We extend this generating function to the boson polynomials of representations of U (n) of the form 〈 (m₁m₂0⋅⋅⋅0) 〉. By considering these polynomials from a different viewpoint, we are able to obtain an explicit decomposition of the Kronecker product of n−1 representations of SU (2).