The Representations and Tensor Operators of the Unitary Groups U(n)

Abstract

This chapter reviews the representation theory of the unitary groups, U(n), deriving a system of basis vectors—the Gel'fand basis—and the matrix elements for the infinitesimal operators that span the Lie algebras of these groups. The chapter discusses the theory of tensor operators defined on the unitary groups (2a-f). The unitary groups, most particularly SU(3) and SU(6), have recently become objects of interest to physicists because of their usefulness in the study of elementary particle symmetries. The U(n) groups, as a family, have a further importance in that all of the classical groups can be embedded as subgroups; this property is much more useful for Lie groups than the corresponding embedding of all finite groups in the symmetric group, S(n). The chapter first gives an informal proof—due to Wigner and Stone—of the Peter-Weyl theorem, which has the corollary that all the irreducible unitary representations of a compact matrix group are generated by Kronecker products of a single faithful representation of the group, which, for convenience, is taken to be the defining representation, that is, the set of all n × n unitary matrices in the case of U(n),…. The problem of the reduction of the Kronecker products of the defining representation is then solved with the assistance of the representation theory of the finite symmetric groups.