From canonical tensor operators of SU(3) and uq(3) to bi-orthogonal coupling coefficients: Explicit expansion
Abstract
Expansion of the matrix elements of SU(3) and uq(3) canonical tensor operators in terms of the bi-orthogonal coupling coefficients and overlaps of the Draayer-Akiyama construction are considered. Special bi-orthogonal extremal isoscalar factors (with subscripts as multiplicity labels and proportional to q-Racah (q-6j) coefficients or the generalized Wilson-Rahman rational bi-orthogonal functions in terms of balanced 4F3(1) or 4φ3(q) hypergeometric series) are used as a natural basis for extremal matrix elements of the highest weight component of the canonical tensor operators of SU(3) in the generating function approach of Biedenharn, Lohe and Louck. The expansion that is obtained (triple sum), together with previously derived asymmetric seed isofactors and elementary overlaps, gives the explicit overlap coefficients and can be used to derive SU(3) and uq(3) canonical tensor operators as well as new explicit normalized seed isofactors with Regge-type symmetry, specified for the minimal null space case.