Characters of Finite Groups
Assume that \(G\) is a finite group of exponent \(m\) with \(k\) conjugacy classes of elements. The operators discussed here are concerned with the ring of class functions on \(G\), defined to be the ring of complex-valued functions on \(G\) that are constant on conjugacy classes. This ring is made into a \({\mathbb{C}}\)-algebra by identifying \(c\in {\mathbb{C}}\) with the constant function that is \(c\) everywhere. In fact we will restrict ourselves to functions with values that are elements of cyclotomic fields. Elements of the ring, that is, objects of type AlgChtrElt, are represented by the \(k\) values (elements of some cyclotomic field \({\mathbb{Q}}(\zeta_n)\)) on the classes. The numbering of those elements matches the numbering of the classes as returned by Classes applied to the underlying group G: Thus X[i] is the value of the character X on the \(i\)-th class, i.e. Classes(G)[i].
- Creation Functions
- Structure Creation
- Element Creation
elt< R | a₁, ..., aₖ :parameters>: AlgChtr, FldCycElt, ..., FldCycElt → AlgChtrEltR ! [ a₁, ..., aₖ ]: AlgChtr, SeqEnum → AlgChtrEltR ! a: AlgChtr, RngIntElt → AlgChtrEltR ! a: AlgChtr, FldRatElt → AlgChtrEltR ! a: AlgChtr, FldCycElt → AlgChtrEltId(R): AlgChtr → AlgChtrEltIdentity(R): AlgChtr → AlgChtrEltOne(R): AlgChtr → AlgChtrEltPrincipalCharacter(G): Grp → AlgChtrEltZero(R): AlgChtr → AlgChtrElt
- The Table of Irreducible Characters
KnownIrreducibles(R): AlgChtr → SeqEnumKnownIrreducibles(G): Grp → SeqEnumCharacterTable(G :parameters): Grp → SeqEnumCharacterTableDS(G :parameters): Grp → SeqEnum, SeqEnumBasis(R): AlgChtr → SeqEnumCharacterTableConlon(G): Grp → SeqEnumLinearCharacters(G): Grp → SeqEnumCharacterDegrees(G): GrpPerm → SeqEnumCharacterDegrees(G): GrpPC → SeqEnumCharacterDegrees(G): GrpMat → SeqEnumCharacterDegrees(G): GrpAb → SeqEnumCharacterDegrees(G, z, p): GrpPC, GrpPCElt, RngIntElt → SeqEnumCharacterDegreesPGroup(G): GrpPC → SeqEnumRationalCharacterTable(G): GrpFin → SeqEnumSaveCharacterTable(T, F): SeqEnum[AlgChtrElt], MonStgElt- Example: Printing a Character Table
- Example: Character Table of an Almost Simple Group
- Character Ring Operations
- Related Structures
Parent(R): AlgChtr → PowCategory(R): AlgChtr → CatNumberOfClasses(R): AlgChtr → RngIntEltNclasses(R): AlgChtr → RngIntEltClassesData(R): AlgChtr → SeqEnum[Tup]Group(R): AlgChtr → GrpPowerMap(R): AlgChtr → MapKernel(x): AlgChtrElt → GrpCentre(x): AlgChtrElt → GrpCoefficientField(x): AlgChtrElt → RngCharacterField(x): AlgChtrElt → RngDegreeOfCharacterField(x): AlgChtrElt → RngIntElt
- Related Structures
- Element Operations
- Arithmetic
- Predicates and Booleans
x in y: AlgChtrElt, AlgChtrElt → BoolEltx notin y: AlgChtrElt, AlgChtrElt → BoolElta in F: FldFunElt, FldFun → BoolElta notin F: FldFunElt, FldFun → BoolEltx eq y: AlgChtrElt, AlgChtrElt → BoolEltx ne y: AlgChtrElt, AlgChtrElt → BoolEltIsCharacter(x): AlgChtrElt → BoolEltIsGeneralizedCharacter(x): AlgChtrElt → BoolEltIsIrreducible(x): AlgChtrElt → BoolEltIsLinear(x): AlgChtrElt → BoolEltIsFaithful(x): AlgChtrElt → BoolEltIsReal(x): AlgChtrElt → BoolEltIsOne(x): AlgChtrElt → BoolEltIsMinusOne(x): AlgChtrElt → BoolEltIsZero(x): AlgChtrElt → BoolEltIsPermutationCharacter(c): AlgChtrElt → BoolElt- Example: Testing for Permutation Characters
- Accessing Class Functions
- Conjugation of Class Functions
x ^ g: AlgChtrElt, GrpElt → AlgChtrEltx ^ H: AlgChtrElt, Grp → { AlgChtrElt }GaloisConjugate(x, j): AlgChtrElt, RngIntElt → AlgChtrEltGaloisOrbit(x): AlgChtrElt → { AlgChtrElt }IsConjugate(x, y): AlgChtrElt, AlgChtrElt → BoolElt, RngIntEltClassPowerCharacter(x, j): AlgChtrElt, RngIntElt → AlgChtrElt
- Functions Returning a Scalar
Degree(x): AlgChtrElt → RngIntEltInnerProduct(x, y): AlgChtrElt, AlgChtrElt → FldCycEltOrder(x): AlgChtrElt → RngIntEltNorm(x): AlgChtrElt → FldCycEltSchur(x, k): AlgChtrElt, RngIntElt → FldCycEltIndicator(x): AlgChtrElt → FldCycEltStructureConstant(G, i, j, k): Grp, RngIntElt, RngIntElt, RngIntElt → RngIntElt
- The Schur Index
SchurIndex(x): AlgChtrElt → RngIntEltSchurIndex(x, Q): AlgChtrElt, FldRat → RngIntEltSchurIndex(x, F): AlgChtrElt, FldAlg → RngIntEltSchurIndices(x): AlgChtrElt → SeqEnumSchurIndices(x, Q): AlgChtrElt, FldRat → SeqEnumSchurIndices(x, F): AlgChtrElt, FldAlg → SeqEnumSchurIndices(C, s, F): FldAlg, SeqEnum, FldAlg → SeqEnum- Example: Computing Schur Indices
- Example: Constructing a Character of Schur Index 6
SchurIndexGroup(n: parameters): RngIntElt → GrpPCCharacterWithSchurIndex(n: parameters): RngIntElt → AlgChtrElt. GrpPC- Attribute
- Constructions for Characters
- Induction, Restriction, and Lifting
Induction(x, G): AlgChtrElt, Grp → AlgChtrEltInduction(Q, G): SeqEnum[AlgChtrElt], Grp → SeqEnum[AlgChtrElt]LiftCharacter(c, f, G): AlgChtrElt, Map, Grp → AlgChtrEltLiftCharacters(T, f, G): [AlgChtrElt], Map, Grp → AlgChtrEltRestriction(x, H): AlgChtrElt, Grp → AlgChtrEltRestriction(Q, H): SeqEnum[AlgChtrElt], Grp → SeqEnum[AlgChtrElt]SubgroupFusion(G, H): Grp, Grp → SeqEnum[RngIntElt]
- Symmetrization
Symmetrization(chi, p): AlgChtrElt, [ RngIntElt ] → AlgChtrEltOrthogonalSymmetrization(chi, p): AlgChtrElt, [ RngIntElt ] → AlgChtrEltSymplecticSymmetrization(chi, p): AlgChtrElt, [ RngIntElt ] → AlgChtrElt- Example: Tensor Product Rule for Symmetrizations
- Example: Symplectic and Orthogonal Symmetrizations
IsOrthogonalCharacter(chi): AlgChtrElt → BoolEltIsSymplecticCharacter(chi): AlgChtrElt → BoolElt
- Permutation Character
- Composition and Decomposition
- Finding Irreducibles
- Induction, Restriction, and Lifting
- Brauer Characters
- Structure Creation
- Element Creation
R ! [ a₁, ..., aₖ ]: AlgChtr, SeqEnum → AlgChtrEltId(R): AlgChtr → AlgChtrEltIdentity(R): AlgChtr → AlgChtrEltOne(R): AlgChtr → AlgChtrEltZero(R): AlgChtr → AlgChtrEltBrauerCharacter(x, p): AlgChtrElt, RngIntElt → AlgChtrEltBrauerCharacter(M): ModGrp → AlgChtrEltBrauerCharacterTable(G, p): Chtr → SeqEnum
- Arithmetic
- Induction, Restriction, and Lifting
Induction(x, G): AlgChtrElt, Grp → AlgChtrEltInduction(Q, G): SeqEnum[AlgChtrElt], Grp → SeqEnum[AlgChtrElt]LiftCharacter(c, f, G): AlgChtrElt, Map, Grp → AlgChtrEltLiftCharacters(T, f, G): [AlgChtrElt], Map, Grp → AlgChtrEltRestriction(x, H): AlgChtrElt, Grp → AlgChtrEltRestriction(Q, H): SeqEnum[AlgChtrElt], Grp → SeqEnum[AlgChtrElt]Blocks(T, p): SeqEnum[AlgChtrElt], RngIntElt → SeqEnum, SeqEnumDefectGroup(T, b, p): SeqEnum[AlgChtrElt], SetEnum[RngIntElt], RngIntElt → GrpDefectGroup(x, p): AlgChtrElt, RngIntElt → Grp- Example: 3-Modular Characters of the Higman-Sims Group
- Database of Character Tables
- Bibliography