The Clebsch-Gordon coefficients for U(m/n) Gel'fand basis and isoscalar factors for U(m/n) ⊃ U(m) × U(n) group chain

Abstract

The outer-product reduction coefficients (ORC) of the graded state permutation group are identified with the Clebsch-Gordan coefficients (CGC) for the special Gel'fand basis of the graded unitary group U(m/n), while the CGC for a general Gel'fand basis of U(m/n) can be easily calculated from the ORC. The isoscalar factors (ISF) for U(m+p/n+q) ⊃ U(m/n) × U(p/q) are shown to be identical with the ISF for U(m+p) ⊃ U(m) × U(p), while the ISF for U(m/n) ⊃ U(m) × U(n) are obtainable from those for U(m+n) ⊃ U(m) × U(n) by simply changing all the partitions for U(n) into their conjugates and taking into account the phase changes. A simple method is given for evaluating the ISF from the ORC. The values of all the CGC and ISF are rank independent.