Sébastien Palcoux - cp's OEIS frontend

A371059 Number of conjugacy classes of pairs of commuting elements in the alternating group A_n.

1, 1, 9, 14, 22, 44, 74, 160, 256, 462, 817, 1494, 2543, 4427, 7699, 13352, 22616, 38610, 65052, 110004, 182961, 305007, 503299, 830648, 1356227, 2212790, 3583419, 5790836

Offset: 1

Sébastien Palcoux, Mar 11 2024

The number of conjugacy classes of pairs of commuting elements in a finite group G is the cardinality of the set {c(a,b) | a,b in G and ab=ba} where c(a,b) = {(gag^(-1),gbg^(-1)) | g in G}.
It is equal to the number of conjugacy classes within the centralizers of class representatives of G.
This reformulation was employed in the sequence-generating program.
It is also equal to the rank of the modular fusion category Z(Rep(G)), the Drinfeld center of Rep(G).
These reformulations are explained in the linked MathOverflow posts.

A. Davydov, Bogomolov multiplier, double class-preserving automorphisms, and modular invariants for orbifolds. J. Math. Phys. 55 (2014), no. 9, 092305, 13 pp.

Sébastien Palcoux, Number of conjugacy classes of pairs of commuting elements, MathOverflow.
Sébastien Palcoux, Is the rank of an integral MTC an upper bound for the prime factors of its FPdim?, MathOverflow.

Cf. A000702.

List([1..10],n->Sum(List(ConjugacyClasses(AlternatingGroup(n)),c->NrConjugacyClasses(Centralizer(AlternatingGroup(n),Representative(c))))));

A371037 Orders of almost simple groups.

Original entry on oeis.org

60, 120, 168, 336, 360, 504, 660, 720, 1092, 1320, 1440, 1512, 2184, 2448, 2520, 3420, 4080, 4896, 5040, 5616, 6048, 6072, 6840, 7800, 7920, 8160, 9828, 11232, 12096, 12144, 12180, 14880, 15600, 16320, 19656, 20160, 24360, 25308, 25920, 29120, 29484, 29760, 31200, 32736, 34440

Offset: 1

Sébastien Palcoux, Mar 08 2024

nonn

A group G is almost simple if there exists a (non-abelian) simple group S for which S <= G <= Aut(S).

			For n = 1, 2, 3, 4 the values a(n) = 60, 120, 168, 336 correspond to the groups A5, S5, PSL(2,7), PGL(2,7), respectively.

Sébastien Palcoux, Table of n, a(n) for n = 1..113
T. Connor and D. Leemans, An atlas of subgroup lattices of finite almost simple groups.
GroupNames, Almost simple groups.
Groupprops, Almost simple group.
Wikipedia, Almost simple group.

Cf. A001034.

m := 100000;;
L := [];;
it := SimpleGroupsIterator(2, m);;
for g in it do
    ag := AutomorphismGroup(g);;
    iag := InnerAutomorphismsAutomorphismGroup(ag);;
    Inter := IntermediateSubgroups(ag, iag).subgroups;;
    LL := [Order(ag), Order(iag)];;
    for h in Inter do
        Add(LL, Order(h));;
    od;
    for o in LL do
        if o <= m and (not o in L) then
            Add(L, o);;
        fi;
    od;
od;
Sort(L);;
Print(L);;

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

Original entry on oeis.org

574, 7315, 63436, 65971, 68587, 90590, 113310, 310730, 311343, 494102, 532159, 585123, 1012810, 1043710, 1107139, 1152907, 1185558, 1343202, 1411338, 1419779, 1425114, 1483682, 1745610, 1898038, 1916226, 2112179, 2161715, 2175315, 2630642, 2753395, 2898555

Offset: 1

Sébastien Palcoux, Jan 27 2024

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.

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).

Winfried Bruns and Sébastien Palcoux, Table of n, a(n) for n = 1..298
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.

Cf. A348305, A001034.

# 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))

Terms a(13) and beyond from Sébastien Palcoux, Dec 30 2024

A354477 Number of fusion rings of multiplicity 4 and rank n.

Original entry on oeis.org

0, 1, 6, 45, 134, 872

Offset: 1

Sébastien Palcoux, Jun 02 2022

See A348305 for the basic definitions and references about fusion rings.

The above terms of the sequence were computed by J. Slingerland and G. Vercleyen (see paper in reference below, Table 2 on page 6).

			a(3) = A354471(4), a(4) = A354472(4), a(5) = A354473(4).

J. Slingerland and G. Vercleyen On Low Rank Fusion Rings, arXiv:2205.15637 [math-ph], 2022.

Cf. A348305, A354471, A354472, A354473.

A354476 Number of fusion rings of multiplicity 3 and rank n.

Original entry on oeis.org

0, 1, 4, 24, 82, 384

Offset: 1

Sébastien Palcoux, Jun 02 2022

See A348305 for the basic definitions and references about fusion rings.

The above terms of the sequence were computed by J. Slingerland and G. Vercleyen (see paper in reference below, Table 2 on page 6).

			a(3) = A354471(3), a(4) = A354472(3), a(5) = A354473(3).

J. Slingerland and G. Vercleyen On Low Rank Fusion Rings, arXiv:2205.15637 [math-ph], 2022.

Cf. A348305, A354471, A354472, A354473.

A354475 Number of fusion rings of multiplicity 2 and rank n.

Original entry on oeis.org

0, 1, 3, 17, 37, 154, 319

Offset: 1

Sébastien Palcoux, Jun 02 2022

See A348305 for the basic definitions and references about fusion rings.

The above terms of the sequence were computed by J. Slingerland and G. Vercleyen (see paper in reference below, Table 2 on page 6).

			a(3) = A354471(2), a(4) = A354472(2), a(5) = A354473(2).

J. Slingerland and G. Vercleyen On Low Rank Fusion Rings, arXiv:2205.15637 [math-ph], 2022.

Cf. A348305, A354471, A354472, A354473.

A354473 Number of fusion rings of rank 5 and multiplicity n.

Original entry on oeis.org

16, 37, 82, 134, 209, 336, 477, 733, 1463, 1794, 2283, 3049

Offset: 1

Sébastien Palcoux, Jun 02 2022

See A348305 for the basic definitions and references about fusion rings.

The above terms of the sequence were computed by J. Slingerland and G. Vercleyen (see paper in reference below, Table 2 on page 6).

			a(1) = A348305(5).

J. Slingerland and G. Vercleyen On Low Rank Fusion Rings, arXiv:2205.15637 [math-ph], 2022.

Cf. A348305.

A354472 Number of fusion rings of rank 4 and multiplicity n.

Original entry on oeis.org

10, 17, 24, 45, 55, 81, 92, 137, 151, 186, 238, 291, 246, 340, 349, 525

Offset: 1

Sébastien Palcoux, Jun 02 2022

See A348305 for the basic definitions and references about fusion rings.

The above terms of the sequence were computed by J. Slingerland and G. Vercleyen (see paper in reference below, Table 2 on page 6).

			a(1)=A348305(4)

J. Slingerland and G. Vercleyen On Low Rank Fusion Rings, arXiv:2205.15637 [math-ph], 2022.

Cf. A348305.

A354471 Number of fusion rings of rank 3 and multiplicity n.

Original entry on oeis.org

4, 3, 4, 6, 5, 9, 6, 10, 12, 9, 10, 20, 9, 13, 16, 25

Offset: 1

Sébastien Palcoux, Jun 01 2022

See A348305 for the basic definitions and references about fusion rings.

The above terms of the sequence were computed by J. Slingerland and G. Vercleyen (see paper in reference below, Table 2 on page 6).

			a(1) = A348305(3).

J. Slingerland and G. Vercleyen On Low Rank Fusion Rings, arXiv:2205.15637 [math-ph], 2022.

Cf. A348305.

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

Original entry on oeis.org

1, 2, 4, 9, 10, 21

Offset: 1

Sébastien Palcoux, Mar 19 2022

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.

			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.

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.

Cf. A000001, A348305.

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A371059 Number of conjugacy classes of pairs of commuting elements in the alternating group A_n.

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A371037 Orders of almost simple groups.

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

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A354477 Number of fusion rings of multiplicity 4 and rank n.

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A354476 Number of fusion rings of multiplicity 3 and rank n.

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A354475 Number of fusion rings of multiplicity 2 and rank n.

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A354473 Number of fusion rings of rank 5 and multiplicity n.

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A354472 Number of fusion rings of rank 4 and multiplicity n.

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A354471 Number of fusion rings of rank 3 and multiplicity n.

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