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Number of complex Grothendieck rings of multiplicity one and rank n.
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%I #15 Jun 06 2022 06:22:36

%S 1,2,4,9,10,21

%N Number of complex Grothendieck rings of multiplicity one and rank n.

%C A complex Grothendieck ring is a fusion ring admitting a categorification into a fusion category over the complex field.

%C See the comments in A348305 for the definition of fusion ring, rank, multiplicity.

%C A complex fusion category is a C-linear semisimple rigid tensor category with finitely many simple objects and finite dimensional spaces of morphisms, such that the neutral object is simple, see the book by Etingof-Gelaki-Nikshych-Ostrik mentioned below.

%C This counting comes from the paper by Liu-Palcoux-Ren mentioned below.

%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.

%e For n=1, there is only the trivial one, so a(1)=1.

%e For n=2, there are only the cyclic group C2 one and the Yang-Lee one, so a(2)=2.

%Y Cf. A000001, A348305.

%K nonn,hard,more

%O 1,2

%A _Sébastien Palcoux_, Mar 19 2022