cp's OEIS Frontend

A369625 Frobenius-Perron dimensions of simple integral fusion rings of rank 4.

%I A369625 #77 Aug 22 2025 19:31:00
%S A369625 574,7315,63436,65971,68587,90590,113310,310730,311343,494102,532159,
%T A369625 585123,1012810,1043710,1107139,1152907,1185558,1343202,1411338,
%U A369625 1419779,1425114,1483682,1745610,1898038,1916226,2112179,2161715,2175315,2630642,2753395,2898555
%N A369625 Frobenius-Perron dimensions of simple integral fusion rings of rank 4.
%C A369625 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.
%C A369625 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.
%C A369625 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.
%C A369625 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.
%C A369625 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.
%C A369625 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.
%D A369625 W. Bruns and S. Palcoux, Classifying simple integral fusion rings, work in progress.
%D A369625 P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Mathematical Surveys and Monographs Volume 205 (2015).
%H A369625 Winfried Bruns and Sébastien Palcoux, <a href="/A369625/b369625.txt">Table of n, a(n) for n = 1..298</a>
%H A369625 M. A. Alekseyev, W. Bruns, S. Palcoux, and F. V. Petrov, <a href="https://arxiv.org/abs/2302.01613">Classification of integral modular data up to rank 13</a>, arXiv:2302.01613 [math.QA], 2023-2024.
%H A369625 W. Bruns, B. Ichim, C. Söger and U. von der Ohe, <a href="https://www.normaliz.uni-osnabrueck.de">Normaliz. Algorithms for rational cones and affine monoids</a>. See also <a href="https://doi.org/10.1016/j.jalgebra.2010.01.031">the paper</a> in J. Algebra 324 (2010), no. 5, 1098--1113.
%H A369625 S. Palcoux, <a href="https://bimsa.net/doc/notes/23787.pdf">Exotic Integral Quantum Symmetry</a>, Slides.
%H A369625 S. Palcoux, <a href="https://youtu.be/77KHe-sL_XU">Why study simple integral fusion categories?</a>, YouTube video.
%H A369625 S. Palcoux, <a href="https://youtu.be/BWKASz72iv8">TPE&Cat Semester 1 Session 23</a>, YouTube video.
%H A369625 S. Palcoux, <a href="https://youtu.be/fOO0hXOgajc">TPE&Cat Semester 1 Session 24</a>, YouTube video.
%H A369625 S. Palcoux, <a href="https://mathoverflow.net/q/463008/34538">Are there infinitely many simple integral fusion rings of rank 4?</a>, MathOverflow, 2024-01-28.
%o A369625 (Python)
%o A369625 # requires Normaliz from version 3.10.2
%o A369625 import math
%o A369625 import PyNormaliz
%o A369625 from PyNormaliz import *
%o A369625 NmzSetNumberOfNormalizThreads(1)
%o A369625 def function(N):
%o A369625     L = []
%o A369625     sN1 = math.isqrt(N//3)
%o A369625     sN = math.isqrt(N)
%o A369625     for i1 in range(3, sN1):
%o A369625         m1 = min(sN, N - i1**2, i1**2 + 1)
%o A369625         for i2 in range(i1+1, m1):
%o A369625             m2 = min(sN, N - i1**2 - i2**2, i2**2 + 1)
%o A369625             for i3 in range(i2+1, m2):
%o A369625                 n = 1 + i1**2 + i2**2 + i3**2
%o A369625                 if n <= N:
%o A369625                     C = Cone(fusion_type = [[1,i1,i2,i3]])
%o A369625                     l = C.FusionRings()
%o A369625                     if len(l)>0:
%o A369625                         L.append(n)
%o A369625     L.sort()
%o A369625     return(L)
%o A369625 print(function(1000))
%Y A369625 Cf. A348305, A001034.
%K A369625 nonn,hard,changed
%O A369625 1,1
%A A369625 _Sébastien Palcoux_, Jan 27 2024
%E A369625 Terms a(13) and beyond from _Sébastien Palcoux_, Dec 30 2024