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Definition of a fusion ring

The notion of fusion ring slightly augments the notion of finite group. Few basics are recalled here. Combinatorially, it is just given by a finite set {1,2,...,n} with an involution i->i^*, and nonnegative integers N_{i,j}^k such that:

• \sum_s N_{i,j}^s N_{s,k}^t = \sum_s N_{j,k}^s N_{i,s}^t (Associativity)

• N_{1,i}^j = N_{i,1}^j = \delta_{i,j} (Unit)

• N_{i^*,k}^{1} = N_{k,i^*}^{1} = \delta_{i,k} (Dual)

• N_{i,j}^k = N_{i^*,k}^j = N_{k,j^*}^i (Frobenius reciprocity)

The multiplicity is the max of (N_{i,j}^k). The rank is n. 

Simple integral 1-Frobenius fusion rings

(Work in progress with Winfried Bruns, updated on 27/12/2024)
Classification of all the non-pointed simple integral 1-Frobenius fusion rings, under the following bounds:
Rank<=r and FPdim<d with (r,d) = (4, 10^15), (5, 10^7), (6, 10^6), (7, 10^5), (8, 20000), (9, 10000), (10, 5000), (11, 3000), (12,1000).

• In bounds: 505 ones (pdf)

Interpolated simple integral fusion rings of Lie type

It should be possible to extend the family of Grothendieck rings of Rep(G(q)), with G(q) a finite (simple) group of Lie type over the finite field of order q (prime power), to a family of interpolated fusion ring "Rep(G(n))" for every positive integer n.

Remark: In the notation "Rep(G(n))", the integer n corresponds to a virtual "finite field of order n", in the same flavor that the non-commutative geometry or the field with one element. In particular, for G classical, it is NOT given by G(Z/nZ), because G(q) is already different from G(Z/qZ) when q=p^r, with p prime and r>1.

Let us consider the interpolated fusion ring "Rep(PSL(2,n))":

• If n ≡ 0 (mod 2), it is of rank n+1, FPdim n(n² -1) and type [[1,1],[n-1,n/2],[n,1],[n+1,(n-2)/2]]. Example: Rep(PSL(2,6)).pdf,
  which is the simple integral fusion ring of rank 7, FPdim 210 and multiplicity 2, found in above classification.

• If n ≡ 1 (mod 2), it is of rank (n+5)/2 and FPdim n(n² -1)/2,

  • if n ≡ 1 (mod 4), it is of type [[1,1],[(n+1)/2,2],[n-1,(n-1)/4],[n,1],[n+1,(n-5)/4],

  • if n ≡ 3 (mod 4), it is of type  [[1,1],[(n-1)/2,2],[n-1,(n-3)/4],[n,1],[n+1,(n-3)/4]. Example: Rep(PSL(2,15)).pdf.

Isotype variations of group-like simple integral fusion rings

The group-like simple integral fusion rings are the Grothendieck rings of Rep(G) with G non-abelian simple group.  

Isotype means of same type. Note that rank=class number and  FPdim=order of G. 

• Rank <= 21, FPdim < 11000000 and group-like: 31 simple ones (pdf),

• Rank <= 10, FPdim < 30000 and (some) isotype variations of group-like: 1554 simple ones (pdf).

It is not a classification of all the isotype variations, but those of a specific kind.

Note that the non-trivial variations above are simple integral fusion rings and are not group-like. 

Simple integral fusion rings not 1-Frobenius

Exploration bounds: Rank<=r and FPdim<d with (r,d) = (7, 1520), (8, 636), (9, 204).

• In bounds: 46 simple and 82 perfect non-simple ones (pdf),

• Out of the bounds: 25 perfect non-simple ones (pdf).

The perfect non-simple ones are just extra, the classification is not complete for them.

Fusion rings of small multiplicity and small rank

• Multiplicity 1 and rank <= 7 : 1, 2, 4, 10, 16, 39, 43 ones of rank 1, 2, 3, 4, 5, 6, 7 (txt), 

• Multiplicity 2 and rank <= 6 : 0, 1, 3, 17, 37, (>)34 ones of rank 1, 2, 3, 4, 5, 6 (pdf),

• Multiplicity 3 and rank <= 5 : 0, 1, 4, 24, (>)81 ones of rank 1, 2, 3, 4, 5 (pdf),

• Multiplicity 4 and rank <= 5 : 0, 1, 6, 45, (>)93 ones of rank 1, 2, 3, 4, 5 (pdf). 

See also: AnyonWiki, Gert Vercleyen, Joost Slingerland, On Low Rank Fusion Rings, https://doi.org/10.1063/5.0148848, arXiv:2205.15637

SageMath codes for classifications and criterions

The end of each above list contains the result of the application of the following criterions. 

• Classifying the fusion rings of small multiplicity and small rank (pdf), 

• Classifying the simple (and perfect) integral fusion rings of Frobenius type (pdf), 

• Classifying the isotype variations of a specific kind (pdf),

• Classifying the simple (and perfect) integral fusion rings not of Frobenius type (pdf), 

• Commutative Schur Product Criterion and Ostrik inequality (pdf), 

• Extended Cyclotomic Criterion and Frobenius type test (pdf).