Recursive calculation of nonprimitive coupling and recoupling brakets

Abstract

The concept of coupling coefficients of angular momentum for the rotation group chain [Formula: see text] can be extended to representations of any group-chain factorisation by defining the generalised notion of a braket. We give a unified approach to recoupling coefficients (rccs), vector-coupling coefficients (vccs), and 3j phases for all group-chain transformations. Category theory is the appropriate tool for studying the representations of groups and algebras. The explicit use of category theory leads to a new recursion scheme for the calculation of brakets. We derive specialisations for calculating nonprimitive rccs and nonprimitive vccs from primitive rccs, primitive vccs, and 3j phases. This new recursion scheme forms the algorithmic core of Racah v4. Racah v4 is a software package developed at the University of Canterbury to calculate group representation coefficients (brakets). PACS Nos.: 02.20.Mp, 03.65.Fd, 31.15.–p, 31.15.Hz, 02.10.Ws