On a solution of the U(N)⊃O(N) state labelling problem for two-rowed representations

Abstract

A discussion of the number of labelling operators specifying an abstract basis for an irreducible representation is given which indicates that in the U(N) contains/implies O(N) state labelling problem, for two-rowed irreducible representations (p,q,0,...,0) of U(N), a single additional commuting labelling operator Lambda is required. An analysis of the tensor representations leads to a suitable labelling scheme involving an additional whole number label lambda . A U(N) contains/implies O(N) branching theorem, for two-rowed representations, is formulated, and the branching multiplicities written down. Combinatorial techniques developed by Green and Bracken in 1973 for the case of U(3) contains/implies SO(3), adapted to the case of U(N) contains/implies O(N), lead to a polynomial identity of the form P( Lambda , Phi ,...)=0, which implicitly defines the labelling operator Lambda (with eigenvalue lambda ) in terms of certain O(N) invariants Phi (which are functions of the U(N) generators) and other known labelling operators and invariants. The calculations also give a cubic polynomial identity, satisfied by the N*N matrix of U(N) generators in two-rowed representations. Some physical applications of the U(N) contains/implies O(N) state labelling problem are briefly mentioned.