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We show how the fusion rules for an affine Kac-Moody Lie algebra g of type A_{n-1}, n = 2 or 3, for all positive integral level k, can be obtained from elementary group theory. The orbits of the kth symmetric group, S_k, acting on k-tuples of integers modulo n, Z_n^k, are in one-to-one correspondence with a basis of the level k fusion algebra for g. If [a],[b],[c] are any three orbits, then S_k acts on T([a],[b],[c]) = {(x,y,z)in [a]x[b]x[c] such that x+y+z=0}, which decomposes into a finite number, M([a],[b],[c]), of orbits under that action. Let N = N([a],[b],[c]) denote the fusion coefficient associated with that triple of elements of the fusion algebra. For n = 2 we prove that M([a],[b],[c]) = N, and for n = 3 we prove that M([a],[b],[c]) = N(N+1)/2. This extends previous work on the fusion rules of the Virasoro minimal models [Akman, Feingold, Weiner, Minimal model fusion rules from 2-groups, Letters in Math. Phys. 40 (1997), 159-169].
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We present an algorithm for an efficient calculation of the fusion rules of twisted representations of untwisted affine Lie algebras. These fusion rules appear in WZW orbifold theories and as annulus coefficients in boundary WZW theories; they provide NIM-reps of the WZW fusion rules.
This is an expository introduction to fusion rules for affine Kac-Moody algebras, with major focus on the algorithmic aspects of their computation and the relationship with tensor product decompositions. Many explicit examples are included with figures illustrating the rank 2 cases. New results relating fusion coefficients to tensor product coefficients are proved, and a conjecture is given which shows that the Frenkel-Zhu affine fusion rule theorem can be seen as a beautiful generalization of the Parasarathy-Ranga Rao-Varadarajan tensor product theorem. Previous work of the author and collaborators on a different approach to fusion rules from elementary group theory is also explained.
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