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COMMENTS
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The notion of fusion ring was introduced by G. Lusztig (1987). See the modern definition on page 60 of the book "Tensor Categories" (2015), in reference below.
It can be seen as a generalization of both finite group (based ring) and its representation (based) ring.
Combinatorially, it is just given by a finite set {1,...,n}, a bijection i->i^* (called dual map) and fusion coefficients N_{i,j}^k which are nonnegative integers satisfying the following axioms:
- Associativity: Sum_s N_{i,j}^s N_{s,k}^t = Sum_s N_{j,k}^s N_{i,s}^t,
- Neutral: N_{1,i}^j = N_{i,1}^j = delta_{i,j},
- Dual: N_{i^*,k}^{1} = N_{k,i^*}^{1} = delta_{i,k},
- Frobenius reciprocity: N_{i,j}^k = N_{i^*,k}^j = N_{k,j^*}^i.
The rank is just n. The multiplicity is the max of (N_{i,j}^k).
The fusion rings are considered up to equivalence.
There is a distinct fusion ring of multiplicity one and rank n for each finite group of order n (just take the group as finite set, the inverse as dual map and N_{g,h}^k = delta_{gh,k}), so a(n)>=A000001(n). The inequality is strict for n>1.
The above terms of the sequence were computed by J. Slingerland and G. Vercleyen (see paper in link below, Table 2 on page 6).
The six first terms were also computed independently by Z. Liu, S. Palcoux and Y. Ren (see link below, page 1).
The (optimized) code computing these terms may be too long to be put here.
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LINKS
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P. Etingof, S. Gelaki, D. Nikshych and V. Ostrik, Tensor Categories, Mathematical Surveys and Monographs Volume 205 (2015).
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