OFFSET
1,1
COMMENTS
See A348305 for the basic definitions and references about fusion rings, or page 60 in Etingof et al. The "rank" of a fusion ring is the cardinal of its basis. The Frobenius-Perron dimension (FPdim) of a fusion ring is the sum of the square of the FPdim of its basic elements.
A fusion ring is called "integral" if the FPdim of its basic elements are integers. The group ring ZG is an example of integral fusion ring, where the finite group G is the basis. The character ring ch(G) is another example of integral fusion ring, where the basis is the set of irreducible characters.
A fusion ring is called "simple" if it has no proper nontrivial fusion subring. The fusion ring ch(G) is simple iff the finite group G is simple.
The fusion ring ch(G) remembers the simple group G (not true for non-simple groups, e.g., D4 and Q8), moreover there are plenty of simple integral fusion rings not of the form ch(G). So a classification of simple integral fusion rings would "really" extend CFSG.
The minimal rank for a non-pointed simple integral fusion ring of the form ch(G) is 5, given by G=A5. But in general, the minimal rank for a simple integral fusion ring is 4, as proved in Alekseyev et al.
The list of twelve simple integral fusion rings of rank 4 and FPdim < 10^6 is available in slide 19 of the talk "Exotic Integral Quantum Symmetry" in the Links section.
REFERENCES
W. Bruns and S. Palcoux, Classifying simple integral fusion rings, work in progress.
P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Mathematical Surveys and Monographs Volume 205 (2015).
LINKS
Winfried Bruns and Sébastien Palcoux, Table of n, a(n) for n = 1..66
M. A. Alekseyev, W. Bruns, S. Palcoux, and F. V. Petrov, Classification of integral modular data up to rank 13, arXiv:2302.01613 [math.QA], 2023-2024.
W. Bruns, B. Ichim, C. Söger and U. von der Ohe, Normaliz. Algorithms for rational cones and affine monoids. See also the paper in J. Algebra 324 (2010), no. 5, 1098--1113.
S. Palcoux, Exotic Integral Quantum Symmetry, Slides.
S. Palcoux, Why study simple integral fusion categories?, YouTube video.
S. Palcoux, TPE&Cat Semester 1 Session 23, YouTube video.
S. Palcoux, TPE&Cat Semester 1 Session 24, YouTube video.
S. Palcoux, Are there infinitely many simple integral fusion rings of rank 4?, MathOverflow, 2024-01-28.
PROG
(Python)
# requires Normaliz from version 3.10.2
import math
import PyNormaliz
from PyNormaliz import *
NmzSetNumberOfNormalizThreads(1)
def function(N):
L = []
sN1 = math.isqrt(N//3)
sN = math.isqrt(N)
for i1 in range(3, sN1):
m1 = min(sN, N - i1**2, i1**2 + 1)
for i2 in range(i1+1, m1):
m2 = min(sN, N - i1**2 - i2**2, i2**2 + 1)
for i3 in range(i2+1, m2):
n = 1 + i1**2 + i2**2 + i3**2
if n <= N:
C = Cone(fusion_type = [[1, i1, i2, i3]])
l = C.FusionRings()
if len(l)>0:
L.append(n)
L.sort()
return(L)
print(function(1000))
CROSSREFS
KEYWORD
nonn,hard
AUTHOR
Sébastien Palcoux, Jan 27 2024
EXTENSIONS
Terms a(13) and beyond from Sébastien Palcoux, Dec 30 2024
STATUS
approved