ClassSum

The $k$-th class sum: add up one conjugacy class. Class sums are the basis of the CENTRE of $k[G]$ — a commutative subalgebra of a usually non-commutative algebra.

WikipediaConjugacy class↗Conjugacy class↗MathWorldConjugacyClass↗WikidataQ1353233↗nLabconjugacy class↗

ClassSum(group, k)the sum of the $k$-th conjugacy class (1-indexed)

SageMathenumeratio_class_sum(group, k)Oscarenumeratio_class_sum(group, k)

IsCentral(group, element)whether an element commutes with everything

SageMathenumeratio_class_sum(group, element)Oscarenumeratio_class_sum(group, element)

Typed signatures

Type	Package	Applies
(expression<CyclicGroup> | expression<DihedralGroup> | expression<GroupDirectProduct>, integer) -> expression	groupalgebra	always

Domain: Group algebras

Details

• A single non-central element is not central, but the class sum containing it always is — that is the point
• Class sums have disjoint supports, so they are linearly independent and form a basis of the centre
• The first class is always the identity alone
• $\dim Z(k[G])$ is the number of conjugacy classes, which is also the number of irreducible characters of $G$

Examples

Basic #

the identity's class is a singleton

live ↗

In$\mathrm{ClassSum}(\mathrm{DihedralGroup}(3), 1)$Out$\mathrm{GroupBasis}(\text{0})$

Scope #

in an abelian group everything is central

live ↗

In$\mathrm{IsCentral}(\mathrm{CyclicGroup}(6), \mathrm{GroupBasis}(\text{3}))$Out$\mathrm{True}$

Properties #

class sums are central

live ↗

In$\mathrm{IsCentral}(\mathrm{DihedralGroup}(3), \mathrm{ClassSum}(\mathrm{DihedralGroup}(3), 2))$Out$\mathrm{True}$

a lone reflection is not

live ↗

In$\mathrm{IsCentral}(\mathrm{DihedralGroup}(3), \mathrm{GroupBasis}(\text{s0}))$Out$\mathrm{False}$

About

Examples	4
Checked against	SageMath 4 agreeOscar 4 agree
Crosswalk	Wikipedia ×2MathWorldWikidatanLabSageMathOscar

Implementation

mappedoscarpackages/oracle/src/mappings.ts

Classes are numbered in our order, by each class's smallest element; Oscar's conjugacy_classes order differs, so the helper re-sorts.

mappedsagepackages/oracle/src/mappings.ts

Classes are numbered in our order, by each class's smallest element, as for Oscar.

mappedwolframpackages/oracle/src/mappings.ts

A class sum is an element of our group algebra over our own element labels (GroupBasis); Wolfram has no group algebra and no conjugacy classes, so there is nothing to sum.

See also: ConjugacyClasses, GroupBasis