Combinatorics of 3n-j Coefficients

J.D. Louck, W.Y.C. Chen, and H.W. Galbraith

  Abstract:  The binary coupling theory of the addition of n+1 angular momenta, or , equivalently, of the reduction of n+1 multiple Kronecker products of the unitary irreducible representations of the unitary group SU(2), is reviewed with emphasis on the combinatorial structure. Using labeled binary trees, we give the generating function for the coupled angular-momentum function corresponding to every binary coupling scheme. From this result, we obtain the generating funcion for the coefficients for all recoupling coefficients. We generalize the methods of Schwinger, but formulated in the ring of polynomials. A key concept for recoupling coefficients is the double Pfaffian, Which is closely related to MacMahon's master theorem. Work in progress includes the classification of 3n-j coefficients, the investigation of representations of U(n+1) that arise from its right action on the 2*(n+1) matrix Z of indeterminates whose columns are the two-component spinors associated with the (n+1)-angular-momentum basis function, and related problems.