Characters and Character Tables

Facilities for working with group characters can be found in Chapter Characters of Finite Groups. In this section we repeat a small number of character intrinsics that are used frequently when computing with \(K[G]\)-modules.

Ordinary Characters

In this section the various intrinsics for computing the table of absolutely irreducible complex characters for a finite group are described.

Character(M): ModGrp -> AlgChtrElt

Given a \(K[G]\)-module \(M\) where \(K\) is the field of rationals or a number field, the character for \(M\) over the field \(K\) is returned.

CharacterTable(G): Chtr -> SeqEnum

The table of irreducible complex characters for the group \(G\) is constructed.

SymmetricCharacterTable(n): RngIntElt -> SeqEnum

AlternatingCharacterTable(n): RngIntElt -> SeqEnum

The table of irreducible complex characters for the symmetric (alternating) group of degree \(n\) is constructed.

RationalCharacterTable(G): Chtr -> SeqEnum

The table of irreducible rational characters for the group \(G\) is constructed.

Brauer Characters

In this section the intrinsic for computing the table of absolutely irreducible Brauer characters of a finite group are described.

BrauerCharacterTable(G, p): Chtr -> SeqEnum

Construct the table of irreducible Brauer characters in characteristic \(p\) for the group \(G\). For soluble groups this is deduced from the ordinary character table. For non-soluble groups the absolutely irreducible \(p\)-modular representations are constructed.

Character(M): ModGrp -> AlgChtrElt

Given a \(K[G]\)-module \(M\), where \(K\) is a finite field of characteristic \(p\) a prime, the \(p\)-modular Brauer character of \(M\) is constructed.