Applications of generating function method to the symmetry and the Kronecker products of SU(n) representations

TL;DR

This paper develops a generating function approach to analyze SU(n) representations, deriving conjugate states, invariants of Kronecker products, and applying these to SU(3) coupling, resulting in new formulas and expressions.

Contribution

It introduces a novel generating function method for SU(n) representations, providing explicit formulas for invariants and new expressions for Wigner symbols.

Findings

Derived conjugate states of SU(n) basis.

Formulated invariants of SU(n) Kronecker products.

Produced a new expression for SU(3) Wigner symbols.

Abstract

Using the generating function of SU(n) we find the conjugate state of SU(n) basis and we find in terms of Gel'fand basis of SU(3(n-1)) the representation of the invariants of the Kronecker products of SU(n). We find a formula for the number of the elementary invariants of SU(n). We apply our method to the coupling of SU(3) and we find a new expression of the isoscalar of Wigner symbols ($\lambda 10,\lambda 2 \mu 2; \lambda 3 \mu 3$).