Explicit Decomposition of r-fold Tensor Products of Irreducible Representations of Classical Groups.
by
Tuong Ton-That
Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA
Coauthors: William H. Klink, Department of Physics and Astronomy, University of Iowa, Iowa City, IA 52242, USA

One of the outstanding problems in Particle Physics is "the multiplicity problem". Typically a many-body system, such as a many- electron atom or many-nucleon nucleous, is described by tensor products of one-particle states. In decomposing tensor products of irreducible representations of a compact group, the same irrreducible representation may appear more than once; the problem is to find a canonical way of treating the equivalent representations that occur in this decomposition. If G is a compact classical group we realize both the tensor product and the irreducible representation spaces as subspaces of a Hilbert space, the Bargmann-Segal-Fock space, on which G acts unitarily. We use invariant theory of the classical groups along with generalized Casimir operators to resolve the multiplicity problem. Generalized Casimir operators are operators from the universal enveloping algebra of outer product of r-fold groups G-action that commute with the diagonal inner group G-action and whose eigenvalues(spectra) can be used to label the multiplicity. In connection with the resolution of the multiplicity problem we prove several reciprocity theorems and give a method of contructing Gelfand-Cetlin bases. To diagonalize many-body Hamiltonians it is often convenient to make use of direct sum bases. Clebsch-Gordan and Racah coefficients are the coefficients that connect tensor product and direct sum bases. As a consequence of our resolution of the multiplicity problem, we also obtain algorithms for computing Clebsch-Gordan and Racah coefficients, which can be implemented by computer programs.

Date received: April 30, 2008