Generating-function method for fusion rules

Abstract

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

On some properties of SU(3) fusion coefficients

“…They are denoted N λ , with λ = (λ 1 , λ 2 ) in the Weyl alcove, and satisfy the following conjugation property: N T λ = N T (λ 1 ,λ 2 ) = N (λ 2 ,λ 1 ) = Nλ, withλ = (λ 2 , λ 1 ). In SU(3) or su(3) k , there is a Z 3 grading τ ("triality") on irreps, stemming from the fact that this discrete group is the center of SU (3). We set for the two fundamental weights τ (N (1,0) ) = 1, τ (N (0,1) ) = 2 mod 3 and more generally τ (λ) ≡ τ (N (λ 1 ,λ 2 ) ) := λ 1 + 2λ 2 mod 3 .…”

“…(9) multiplied by t from equ. (8) multiplied by s, we eliminate the unwanted Λ 3 and get (s 3 l 1 − s 2 tG + st 2 G T − t 3 l 1) · X(s, t) = (s 3 − st)Λ 1 (s) + (st − t 3 )Λ 2 (t) .…”

“…The threshold level (or threshold, for short) k min 0 of a triple (or of a branching λ ⊗ µ → ν) is the smallest value of k for which the fusion coefficient N (k) ν λµ is non-zero. It is known that, for any given triple of irreps, the fusion coefficient is an increasing function of the level, and that it becomes equal to its classical value N ν λµ when k reaches a value k max In terms of couplings (or pictographs), the discussion goes as follows: for fixed λ, µ and ν and a given k, we have a set of N From now on, we consider the special case of SU (3). In all cases, as we saw, for any given triple of irreps, the fusion coefficient is equal to 0 when k < k min Adapting the results of [16] to our own notations 9 (see also [4] and the discussion at the end of our section 5.1.3), we have 10 k min 0 (λ, µ; ν) = max λ 1 + λ 2 ,…”

Fusion rules and the Patera-Sharp generating-function method

“…As described in [12], the Hilbert basis theorem guarantees that every solution can be expanded in terms of the elementary solutions of these inequalities. This is a key concept for the following (see [13] for an extensive discussion of these methods). A sum of two solutions translates into the product of the corresponding couplings, more precisely, to the stretched product (denoted by ·) of the corresponding two LR tableaux.…”

“…(For more complicated cases, we point out that powerful methods to find the elementary solutions are described in [10].) These correspond to the following LR tableaux, denoted respectively E 1 , E 2 , E 3 :…”