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

Author

Sébastien Palcoux, Mar 11 2024

Keywords

• nonn
• more

Comments

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.

References

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

Links

• 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.

Crossrefs

Cf. A000702.

Programs

GAP

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