A352506 Number of complex Grothendieck rings of multiplicity one and rank n.

1, 2, 4, 9, 10, 21

Offset: 1

Author

Sébastien Palcoux, Mar 19 2022

Keywords

• nonn
• hard
• more

Comments

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

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

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.

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

Examples

			For n=1, there is only the trivial one, so a(1)=1.
For n=2, there are only the cyclic group C2 one and the Yang-Lee one, so a(2)=2.
		

Links

• P. Etingof, S. Gelaki, D. Nikshych and V. Ostrik, Tensor Categories, Mathematical Surveys and Monographs Volume 205 (2015).
• Z. Liu, S. Palcoux and Y. Ren, Classification of Grothendieck rings of complex fusion categories of multiplicity one up to rank six, Lett Math Phys 112, 54 (2022); arXiv version, arXiv:2010.10264 [math.CT], 2020-2021.

Crossrefs

Cf. A000001, A348305.