A projection based approach to the Clebsch-Gordan multiplicity problem for compact semisimple Lie groups: I. General formalism

Abstract

The authors describe a method, analogous to the Elliott method (1958) for O(3) contained in/implied by U(3), for resolution of the multiplicity in the decomposition into irreducibles of the tensor product of two irreducible representations of a compact semisimple Lie group G. The method is based on a decomposition of highest weight vectors in the product representation into direct product states, focusing on components of the form e(X)e+mu with e a weight vector and e+mu a highest weight vector. The weight of the vector e corresponds to the 'shift' weight in a tensor operator formulation of the problem. A result from an earlier paper (Gould et al. 1984) based on the fact that a tensor product representation can be generated cyclically from the product of a highest and a lowest weight vector is used to give an explicit characterisation of the space of shift weight vectors e that can appear in the decomposition of a highest weight state. This characterisation is in terms of lowering operators in the complexified Lie algebra L of G, and closely parallels Verma's well known enveloping algebra characterisation (1968) of the highest weight states of finite-dimensional irreducible representations of complex semisimple Lie algebra.