Berenstein-Zelevinsky triangles, elementary couplings, and fusion rules

Bégin, Luc; Kirillov, Alexander; Mathieu, Pierre; Walton, Mark A.

doi:10.1007/bf00761494

Cited by 22 publications

(41 citation statements)

References 15 publications

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“…When these inequalities are re-expressed in terms of BZ triangle data, they reproduce the threshold formula presented in [7].…”

mentioning

confidence: 57%

“…This is easily checked to be equivalent to the formula given in [6,7] in terms of BZ triangle data (cf. …”

Section: The Generating Function For Su(3) Fusion Rulesmentioning

confidence: 93%

“…Consider now the construction of the set of fusion elementary couplings using outer- 1), (a, a), (a, 1), (1, a)} (6.6) and this leads respectively to E 1 , E 2 , E 3 , which all have k 0 = 2 and a new elementary [7]). Notice that at the level of tensor-products, F is a composite product C 1 C 2 C 3 .…”

Section: The Su(4) Generating Functionmentioning

confidence: 99%

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Generating-function method for fusion rules

Bégin

Cummins

Mathieu

2000

Journal of Mathematical Physics

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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

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“…When these inequalities are re-expressed in terms of BZ triangle data, they reproduce the threshold formula presented in [7].…”

mentioning

confidence: 57%

“…This is easily checked to be equivalent to the formula given in [6,7] in terms of BZ triangle data (cf. …”

Section: The Generating Function For Su(3) Fusion Rulesmentioning

confidence: 93%

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Generating-function method for fusion rules

Bégin

Cummins

Mathieu

2000

Journal of Mathematical Physics

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“…3 [14], and their formulas are also surprisingly compact. Incidentally, an analogous modular transformation matrix S to (3.2b) exists for the socalled admissible representations of X (1) r at fractional level [86].…”

Section: Examples Of Modular Data and Fusion Ringsmentioning

confidence: 97%

“…to fix the affine algebra and vary k. This idea still hasn't been seriously exploited (but e.g. see 'threshold level' in [13,14]). …”

Section: Examples Of Modular Data and Fusion Ringsmentioning

confidence: 99%

Modular Data: The Algebraic Combinatorics of Conformal Field Theory

Gannon

2005

J Algebr Comb

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Abstract. This paper is primarily intended as an introduction for mathematicians to some of the rich algebraic combinatorics arising in for instance conformal field theory (CFT). It tries to refine, modernise, and bridge the gap between papers [6] and [55]. Our paper is essentially self-contained, apart from some of the background motivation (Section 1) and examples (Section 3) which are included to give the reader a sense of the context. Detailed proofs will appear elsewhere. The theory is still a work-in-progress, and emphasis is given here to several open questions and problems.

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Remarks on the structure constants of the Verlinde algebra associated to sl3

Felder

Varchenko

1996

Letters in Mathematical Physics

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The Verlinde fusion algebra is an associative commutative algebra associated to a Wess-Zumino-Witten model of conformal field theory [V,F,GW,K,S]. Such a model is labelled by a simple Lie algebra g and a natural number k called level. The Verlinde algebra A(g, k) is a finitely generated algebra with generators V λ enumerated by irreducible g-modules admissible for the model. The structure constants N ν λ,µ of the multiplication V λ · V µ = ν N ν λ,µ V ν are non-negative integers important for applications. ( We use the formula in [K, Sec.13.35] as a definition of the structure constants.) Example 1. The algebra A(sl 2 , k) has k+1 generators V 0 , ..., V k . For fixed λ, ν and varying µ, the structure constants N µ+ν λ,µ are either zero or form the characteristic function of an interval with respect to µ. Namely, N µ+ν λ,µ = 0, if λ − ν is odd or if |ν| > λ. If λ−ν is even and |ν| < λ, then N µ+ν λ,µ = 1 for µ ∈ [(λ−ν)/2, k −(λ+ν)/2] and N µ+ν λ,µ = 0 otherwise. It is interesting that after an affine change of the variable the function N µ+ν λ,µ of µ becomes the weight function of the irreducible sl 2 -module with highest weight k − λ.In this paper we give a similar formula for the structure constants of the Verlinde algebra associated to sl 3 . Weight Functions.Let P = Z 3 /Z · (1, 1, 1) be the two dimensional weight lattice of sl 3 . LetFor a natural number k introduce coordinates on P:y 1 (µ) := (α 1 , µ), y 2 (µ) := (α 2 , µ), y 3 (µ) := k + (α 3 , µ) = k − y 1 − y 2 ,where (x, y) = x 1 y 1 + x 2 y 2 + x 3 y 3 .

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Berenstein-Zelevinsky triangles, elementary couplings, and fusion rules

Cited by 22 publications

References 15 publications

Generating-function method for fusion rules

Generating-function method for fusion rules

Modular Data: The Algebraic Combinatorics of Conformal Field Theory

Remarks on the structure constants of the Verlinde algebra associated to sl3

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