Generating-function method for tensor products

TL;DR

This paper introduces a generating-function approach for calculating tensor product fusion rules in affine Lie algebras, offering a more efficient alternative to character methods by using Diophantine equations and Grobner bases.

Contribution

It presents a novel method for tensor product calculations that simplifies the process and overcomes technical difficulties of previous character-based approaches.

Findings

Developed a new approach using linear Diophantine equations.

Applied Grobner bases to find relations among elementary couplings.

Provided an algorithm for identifying forbidden couplings.

Abstract

This is the first of two articles devoted to a exposition of the generating-function method for computing fusion rules in affine Lie algebras. The present paper is entirely devoted to the study of the tensor-product (infinite-level) limit of fusions rules. We start by reviewing Sharp's character method. An alternative approach to the construction of tensor-product generating functions is then presented which overcomes most of the technical difficulties associated with the character method. It is based on the reformulation of the problem of calculating tensor products in terms of the solution of a set of linear and homogeneous Diophantine equations whose elementary solutions represent ``elementary couplings''. Grobner bases provide a tool for generating the complete set of relations between elementary couplings and, most importantly, as an algorithm for specifying a complete, compatible set of ``forbidden couplings''.