If V (m) is an irreducible representation space for the unitary group U(N), then the r-fold tensor product space, V (m)1⊗⋯⊗V(m)r is in general reducible. Such a reducible representation can be reduced to a direct sum of irreducible representation spaces, albeit with multiplicity. Clebsch-Gordan coefficients are the overlap coefficients between an orthonormal basis in the tensor product space and an orthonormal basis in the direct sum space. Since such coefficients are basis dependent, bases in the U(N) irrep spaces must be chosen. In this paper we use the Gelfand-Zetlin basis, built out of the chain of subgroups U(N) ⊃ ⋯ ⊃ U(1). Our first result is to derive algorithms for generating Gelfand-Zetlin bases from Gelfand-Zetlin tableaux. Given these concrete basis realizations we develop algorithms for computing the Clebsch-Gordan coefficients themselves.
- Gelfand-Zetlin basis
- U(N) Clebsch-Gordan coefficients
ASJC Scopus subject areas
- Applied Mathematics