SU(3) Clebsch-Gordan coefficients and some of their symmetries

TL;DR

This paper explores the construction, symmetries, and applications of su(3) Clebsch-Gordan coefficients, emphasizing their Weyl symmetries and relation to su(2) technology, with examples of multiplicity-free decompositions.

Contribution

It introduces a method to analyze su(3) Clebsch-Gordan coefficients using su(2) basis states and Weyl group symmetries, providing new insights and computational techniques.

Findings

Weyl symmetries generalize su(2) m-> -m symmetry

Coefficients can be expressed via su(2) technology

Examples include multiplicity-free decompositions

Abstract

We discuss the construction and symmetries of su(3) Clebsch-Gordan coefficients arising from the su(3) basis states constructed as triple tensor products of two-dimensional harmonic oscillator states. Because of the su(2) symmetry of the basis states, matrix elements and recursion relations are easily expressed in terms of su(2) technology. As the Weyl group has a particularly simple action on these states, Weyl symmetries of the su(3) coupling coefficients generalizing the well known m-> -m symmetry of su(2) coupling can be obtained, so that any coefficient can be obtained as a sum of Weyl-reflected coefficients lying in the dominant Weyl sector. Some important cases of multiplicity-free decomposition are discussed as examples of applications.