Application of double Gel’fand polynomials to the symmetric group and spin–isospin wave functions of cluster systems

Abstract

The theory of double Gel’fand polynomials is applied to irreducible representations of the symmetric and SU4 groups with the aim to treat spin–isospin wave functions of nuclear cluster systems. Multiplicity-free recoupling coefficients of the symmetric group are connected with special types of Clebsch–Gordan coefficients of the unitary group. The standard phase conventions of the Yamanouchi basis and of the multiplicity-free recoupling coefficients are proved to be derivable from natural phase conventions of double Gel’fand polynomials and these special Clebsch–Gordan coefficients. By extending the concept of double Gel’fand polynomials, useful expansion formulas are derived with respect to the determinant associated with a matrix tensor product. A simple example of their application is given for normalization kernels of two-body systems composed of s-shell clusters and for SU4 Clebsch–Gordan coefficients in the spin–isospin representation needed therein.