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Type A fusion rules from elementary group theory | | Alex J. Feingold
; Michael D. Weiner
; | | Date: |
20 Dec 2000 | | Subject: | Quantum Algebra; Mathematical Physics MSC-class: 17B67, 17B65, 81T40 (Primary) 81R10, 05E10 (Secondary) | math.QA hep-th math-ph math.MP | | Abstract: | 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]. | | Source: | arXiv, math.QA/0012194 | | Services: | Forum | Review | PDF | Favorites | |
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