Type A fusion rules from elementary group theory [article]

Alex J. Feingold, Michael D. Weiner
2000 arXiv   pre-print
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)∈ [a]x[b]x[c] such that x+y+z=0, which decomposes into a finite number,
more » ... 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].
arXiv:math/0012194v1 fatcat:5kpr7dnjubhkfi4qqbcsykz4pq