Character Ring Operations

Related Structures

Parent(R): AlgChtr -> Pow

Category(R): AlgChtr -> Cat

NumberOfClasses(R): AlgChtr -> RngIntElt

Nclasses(R): AlgChtr -> RngIntElt

Given the ring \(R\) of class functions, return the number of conjugacy classes of the finite group underlying \(R\).

ClassesData(R): AlgChtr -> SeqEnum[Tup]

Given the ring \(R\) of class functions, return a sequence of pairs, one pair <o,n> for each conjugacy class of the underlying group, where \(o\) is the order of the group elements in the class, and \(n\) is the length of (number of group elements in) the class.

Group(R): AlgChtr -> Grp

Given the ring \(R\) of class functions on a finite group \(G\), return \(G\). This will cause an error if \(R\) does not have a group attached. If unsure whether or not \(R\) has a group attached, check the Group attribute of \(R\) using assigned R`Group.

PowerMap(R): AlgChtr -> Map

Given a character ring \(R\), return the associated class power map. This will cause an error unless \(R\) has a power map assigned, or has a group attached. The PowerMap is an attribute of \(R\), as is Group, and its presence can be checked using assigned R`PowerMap.

If the power map of \(R\) is not already assigned, and there is an assigned group, this function will compute the power map using operations within the group.

Kernel(x): AlgChtrElt -> Grp

The kernel of the character \(x\) of \(G\), i.e. the normal subgroup of \(G\) consisting of those elements \(g\) for which \(x(g) = x(1)\).

Centre(x): AlgChtrElt -> Grp

The centre of the character \(x\) of \(G\), i.e. the subgroup of \(G\) consisting of those classes \(C\) of \(G\) for which \(|x(g)|\), \(g\) in \(C\), is equal to the degree of \(x\).

CoefficientField(x): AlgChtrElt -> Rng

The minimal cyclotomic field containing all values of the class function \(x\).

CharacterField(x): AlgChtrElt -> Rng

The subfield of the coefficient field of the class function \(x\) that is generated by the values of \(x\).

DegreeOfCharacterField(x): AlgChtrElt -> RngIntElt

The degree of the character field of the class function \(x\) as an extension of the rational numbers.