Recoupling coefficients of the general linear group in bases adapted to shell theories

Abstract

Recoupling coefficients for tensor representations of the general linear group Gl(n) are identified with analogous quantities in representations of the symmetric group SN. Two basis labeling schemes in Gl(n) are considered: (a) uses weights and outer product labels from SN, and (b) uses outer plethysms in SN and labels with respect to some elementary subgroup, usually SU(2). Scheme (a) corresponds to a generalized Gel’fand–Tsetlin basis and is the one usually adopted in elementary particle theories. Scheme (b) corresponds to the basis usually adopted in nuclear and atomic shell theory. The transformation between the two equivalent bases is identified with certain weighted double coset matrix elements (WDCME) of SN. Racah factors are generalized isoscalar factors in scheme (a) and have previously been identified with certain WDCME in that basis. In scheme (b) Racah factors determine the coefficients of fractional parentage (CFP) and are here identified with certain double coset matrix elements (DCME) of SN. Identification of these recoupling coefficients with the analogous quantities in SN exposes new symmetries and orthogonality properties of the coefficients which follow from the representation theory of SN. Some particular examples are verified by coefficients evaluated using well established techniques for SU(2).