The Clebsch-Gordan coefficients and isoscalar factors of the graded unitary group SU(m/n)

Abstract

It is pointed out that the Clebsch-Gordan coefficients (CGC) of the permutation group are the coefficients which couple irreducible bases (IRB) of SU(m/n) and SU(p/q) to the IRB of SU(mp+nq/mp+np), and the outer-product reduction coefficients (ORC) of the permutation group are the coefficients which couple the IRB of SU(m/n) and SU(p/q) into the IRB of SU(m+p/n+q). The ORC of permutation groups with due modification in signs are the CGC for the special Gel'fand basis of SU(m/n). The isoscalar factor (ISF) for the permutation group chain S(f) contains/implies S(f1)*S(f2) is the ISF for the graded unitary group chain SU(mp+nq/mq+np) contains/implies SU(m/n)*SU(p/q), and the outer-product ISF for the group chain S(f) contains/implies S(f1)*S(f2) is the ISF for SU(mp+nq/mq+np) contains/implies SU(m/n)(X)SU(p/q). All these coefficients can be calculated easily and some are already available.