Subsections

• Structure Creation
• Element Creation
• The Table of Irreducible Characters

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 known yet. Information about the irreducible characters is stored in the ring when it is computed.

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 conjugating, restricting, inducing etc. existing class functions.

elt< R | a_1, ..., a_k :parameters> : AlgChtr, FldCycElt, ..., FldCycElt -> AlgChtrElt

R ! [ a_1, ..., a_k ] : AlgChtr, Eseq -> 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 := true, then the resulting character is flagged to be a proper character.

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.

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).

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).

The function CharacterTable can be invoked to determine the complete or a partial table of irreducible characters on a finite group G. If necessary, an existing character table can be supplemented by another call to the function. The known irreducible characters are stored in the character ring of G.

The functions in this section return character tables, which are enumerated sequences of characters that are flagged to allow printing in a special format.

KnownIrreducibles(R) : AlgChtr -> SeqEnum

KnownIrreducibles(G) : Grp -> SeqEnum

Given a finite group G, or a class function space R of G, return the table of irreducible characters currently stored. Such a table is a sequence of characters with specially formatted printing.

It should be noted that characters are stored with some information about whether they are at worst proper characters, generalized characters or class functions. From this information, it is often possible to deduce with little effort that a character is, in fact, irreducible. When a new irreducible is found this way, it is immediately inserted into the table of irreducible characters.

CharacterTable(G; parameters) : Grp -> SeqEnum

    ClassMatrices: SeqEnum              Default: []

    ClassMatrixLimit: RngIntElt         Default: Infinity

    MinChars: RngIntElt                 Default: Infinity

Basis(R) : AlgChtr -> SeqEnum

Given a finite group G, construct the table of irreducible characters of G. The conjugacy classes of G will be computed if necessary; if the user wishes to exert control over the computation of the classes, the function Classes can be invoked on G before calling CharacterTable. The characters are found using the Dixon [J.D. Dixon, High-speed computation of group characters, Numerische Mathematik 10 (1967), 446--450]-Schneider [G.J.A. Schneider, Dixon's Character Table Algorithm Revisited, J. Symbolic Computation 9 (1990), 601--606] algorithm.

The function Basis takes the character ring R of G as input and performs the same computation to find the basis for R consisting of the irreducible characters. This function does not have the optional parameters described below.

Both CharacterTable and Basis return a character table, which is a enumerated sequence of elements of the character ring over G (if necessary, the ring is created) that only differs from arbitrary sequences of class functions with respect to printing.

There are three optional parameters for CharacterTable. Setting MinChars := m (taking non-negative integer values m) indicates that the computation may be finished as soon as at least m additional characters have been found. By putting ClassMatrixLimit := n the user can limit the number of class matrices used (n geq1 an integer). It may be that the calculation of all irreducible characters (or at least m of them if MinChars has been set) cannot be completed by using k class matrices, in which case an incomplete character table is returned. The optional argument ClassMatrices (a sequence of integers in the range 1 ... k, where k is the number of conjugacy classes of G) can be used to specify a preference for the order in which class matrices should be used.

Given a finite group G, determine the (partial) character table containing only the linear characters. Such a table is a sequence of characters with specially formatted printing.