Graphical Tensor Product Reduction Scheme for the Lie Algebras so(5) = sp(2), su(3), and g(2)

TL;DR

This paper presents a detailed graphical tensor product reduction scheme for the rank 2 Lie algebras so(5), sp(2), su(3), and g(2), enabling efficient decomposition of tensor products into irreducible representations.

Contribution

It introduces a practical graphical method for tensor product reduction in rank 2 Lie algebras, including detailed landscapes and weight diagrams, along with computer code implementation.

Findings

Efficient graphical reduction scheme for tensor products.

Comprehensive landscapes and weight diagrams provided.

Computer code available for practical calculations.

Abstract

We develop in detail a graphical tensor product reduction scheme, first described by Antoine and Speiser, for the simple rank 2 Lie algebras so(5) = sp(2), su(3), and g(2). This leads to an efficient practical method to reduce tensor products of irreducible representations into sums of such representations. For this purpose, the 2-dimensional weight diagram of a given representation is placed in a "landscape" of irreducible representations. We provide both the landscapes and the weight diagrams for a large number of representations for the three simple rank 2 Lie algebras. We also apply the algebraic "girdle" method, which is much less efficient for calculations by hand for moderately large representations. Computer code for reducing tensor products, based on the graphical method, has been developed as well and is available from the authors upon request.