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%I #53 Jan 04 2024 03:26:42
%S 1,2,4,10,16,39,43,96,142
%N Number of fusion rings of multiplicity one and rank n.
%C 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.
%C It can be seen as a generalization of both finite group (based ring) and its representation (based) ring.
%C 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:
%C - Associativity: Sum_s N_{i,j}^s N_{s,k}^t = Sum_s N_{j,k}^s N_{i,s}^t,
%C - Neutral: N_{1,i}^j = N_{i,1}^j = delta_{i,j},
%C - Dual: N_{i^*,k}^{1} = N_{k,i^*}^{1} = delta_{i,k},
%C - Frobenius reciprocity: N_{i,j}^k = N_{i^*,k}^j = N_{k,j^*}^i.
%C The rank is just n. The multiplicity is the max of (N_{i,j}^k).
%C The fusion rings are considered up to equivalence.
%C 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.
%C The above terms of the sequence were computed by J. Slingerland and G. Vercleyen (see paper in link below, Table 2 on page 6).
%C The six first terms were also computed independently by Z. Liu, S. Palcoux and Y. Ren (see link below, page 1).
%C The (optimized) code computing these terms may be too long to be put here.
%D G. Lusztig, Leading coefficients of character values of Hecke algebras, Proc. Symp. in Pure Math., 47, pp. 235-262 (1987).
%H AnyonWiki, <a href="http://www.thphys.nuim.ie/AnyonWiki/index.php/Fusion_ring">Fusion ring</a>.
%H P. Etingof, S. Gelaki, D. Nikshych and V. Ostrik, <a href="http://www-math.mit.edu/~etingof/egnobookfinal.pdf">Tensor Categories</a>, Mathematical Surveys and Monographs Volume 205 (2015).
%H Z. Liu, S. Palcoux and Y. Ren, <a href="https://doi.org/10.1007/s11005-022-01542-1">Classification of Grothendieck rings of complex fusion categories of multiplicity one up to rank six</a>, Lett Math Phys 112, 54 (2022); <a href="https://arxiv.org/abs/2010.10264">arXiv version</a>, arXiv:2010.10264 [math.CT], 2020-2021.
%H nLab, <a href="https://ncatlab.org/nlab/show/fusion+ring">fusion ring</a>.
%H J. Slingerland and G. Vercleyen, <a href="https://mathpicture.fas.harvard.edu/files/mathpicture/files/harvard_picture_language_10-2020.pdf">Exploring small fusion rings and tensor categories</a>, Harvard Picture Language Seminar, 20th October 2020,
%H J. Slingerland and G. Vercleyen, <a href="https://arxiv.org/abs/2205.15637">On Low Rank Fusion Rings</a>, arXiv:2205.15637 [math-ph], 2022.
%e For n=1, there is only the trivial fusion ring, so a(1)=1.
%e For n=2, there are only the fusion ring of the cyclic group C2 and the Yang-Lee fusion ring, so a(2)=2.
%Y Cf. A000001.
%K nonn,hard,more
%O 1,2
%A _Sébastien Palcoux_, Oct 10 2021
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