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

1, 2, 4, 9, 10, 21

Offset

1,2

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.

Links

Table of n, a(n) for n=1..6.

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.

Example

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.

Crossrefs

Cf. A000001, A348305.

Sequence in context: A103078 A060756 A075347 * A005733 A372915 A096692

Adjacent sequences: A352503 A352504 A352505 * A352507 A352508 A352509

Keyword

nonn,hard,more

Author

Sébastien Palcoux, Mar 19 2022

Status

approved