Creation Functions

Structure Creation

ClassFunctionSpace(G) : Grp -> AlgChtr

CharacterRing(G) : Grp -> AlgChtr

Given a finite group G, create the ring of complex-valued class functions. This function will trigger the computation of the conjugacy classes of G if these are not yet known. Information about the irreducible characters is stored in the ring when it is computed.

ClassFunctionSpace(Q) : SeqEnum -> AlgChtr

CharacterRing(Q) : SeqEnum -> AlgChtr

Construct the ring of complex-valued class functions for a group that has classes data as given in Q. The sequence Q must be a sequence of pairs <o, n>, each representing one conjugacy class, where o is the order of the elements in the class, and n is the length of the class. The first class must be the class of the group identity element. This is a sequence as returned by the ClassesData intrinsic.

This function allows the creation of a space to work with characters without the attached group.

Element Creation

The elementary constructions for class functions are listed. Other useful ways of defining class functions and characters are defined in sections discussing the permutation character, the (de)composition functions, and the sections on the conjugating, restricting and inducing of class functions.

elt< R | a₁, ..., a_k :parameters> : AlgChtr, FldCycElt, ..., FldCycElt -> AlgChtrElt

R ! [ a₁, ..., a_k ] : AlgChtr, SeqEnum -> AlgChtrElt

Given the ring of class functions R of a finite group G with k conjugacy classes and k elements a_i contained in some common cyclotomic field, create a class function on G for which the value on the i-th class is equal to the i-th term a_i.

     Character: BoolElt                  Default: false

If Character is set to true, then the resulting character is flagged to be a proper character.

R ! a : AlgChtr, RngIntElt -> AlgChtrElt

R ! a : AlgChtr, FldRatElt -> AlgChtrElt

R ! a : AlgChtr, FldCycElt -> AlgChtrElt

Define a constant class function for the ring of class functions R of the group G. Here a is allowed to be an integer, a rational field element or a cyclotomic field element.

Id(R) : AlgChtr -> AlgChtrElt

Identity(R) : AlgChtr -> AlgChtrElt

One(R) : AlgChtr -> AlgChtrElt

PrincipalCharacter(G) : Grp -> AlgChtrElt

Given the finite group G or its ring of class functions R, create the principal character (which takes on the value 1 on every element of G).

Zero(R) : AlgChtr -> AlgChtrElt

Given a ring of class functions R create its zero element (which is the class function that takes on the value 0 on every element of the group).