This is the first of two articles devoted to a exposition of the generatingfunction 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". 11/98 (revised 06/99,01/00) 1
A closed and explicit formula for all [Formula: see text] fusion coefficients is presented which, in the limit k→∞, turns into a simple and compact expression for the su(3) tensor product coefficients. The derivation is based on a new diagrammatic method which gives directly both tensor product and fusion coefficients.
Fermionic-type character formulae are presented for charged irreducible modules of the graded parafermionic conformal field theory associated to the coset osp(1, 2) k / u(1). This is obtained by counting the weakly ordered 'partitions' subject to the graded Z k exclusion principle. The bosonic form of the characters is also presented. 01/03
This is the second of two articles devoted to an exposition of the generating-function method for computing fusion rules in affine Lie algebras. The present paper focuses on fusion rules, using the machinery developed for tensor products in the companion article. Although the Kac-Walton algorithm provides a method for constructing a fusion generating function from the corresponding tensor-product generating function, we describe a more powerful approach which starts by first defining the set of fusion elementary couplings from a natural extension of the set of tensor-product elementary couplings.A set of inequalities involving the level are derived from this set using Farkas' lemma.These inequalities, taken in conjunction with the inequalities defining the tensor products, define what we call the fusion basis. Given this basis, the machinery of our previous paper may be applied to construct the fusion generating function. New generating functions for sp(4) and su(4), together with a closed form expression for their threshold levels are presented.05/99, revised 04/00 (arXiv:hepth/0005002) 1
We present a general scheme for describing su(N ) k fusion rules in terms of elementary couplings, using Berenstein-Zelevinsky triangles. A fusion coupling is characterized by its corresponding tensor product coupling (i.e. its Berenstein-Zelevinsky triangle) and the threshold level at which it first appears. We show that a closed expression for this threshold level is encoded in the Berenstein-Zelevinsky triangle and an explicit method to calculate it is presented. In this way a complete solution of su(4) k fusion rules is obtained.
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