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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 C-algebra by identifying c∈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 Q(ζn)) 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].
Acknowledgements
Creation Functions
Structure Creation
Element Creation
The Table of Irreducible Characters
Character Ring Operations
Related Structures
Element Operations
Arithmetic
Predicates and Booleans
Accessing Class Functions
Conjugation of Class Functions
Functions Returning a Scalar
The Schur Index
Attribute
Constructions for Characters
Induction, Restriction and Lifting
Symmetrization
Permutation Character
Composition and Decomposition
Finding Irreducibles
Brauer Characters
Structure Creation
Element Creation
Arithmetic
Induction, Restriction and Lifting
Database of Character Tables
Bibliography
Creation Functions
Structure Creation
ClassFunctionSpace(G) : Grp -> AlgChtr
ClassFunctionSpace(Q) : SeqEnum -> AlgChtr
Element Creation
elt< R | a1, ..., ak :parameters> : AlgChtr, FldCycElt, ..., FldCycElt -> AlgChtrElt
R ! a : AlgChtr, RngIntElt -> AlgChtrElt
Id(R) : AlgChtr -> AlgChtrElt
Zero(R) : AlgChtr -> AlgChtrElt
The Table of Irreducible Characters
KnownIrreducibles(R) : AlgChtr -> SeqEnum
CharacterTable(G :parameters) : Grp -> SeqEnum
CharacterTableDS(G :parameters) : Grp -> SeqEnum, SeqEnum
Basis(R) : AlgChtr -> SeqEnum
CharacterTableConlon(G) : Grp -> SeqEnum
LinearCharacters(G): Grp -> SeqEnum
CharacterDegrees(G): GrpPerm -> SeqEnum
CharacterDegrees(G, z, p): GrpPC, GrpPCElt, RngIntElt -> SeqEnum
CharacterDegreesPGroup(G): GrpPC -> SeqEnum
RationalCharacterTable(G): GrpFin -> SeqEnum
SaveCharacterTable(T, F): SeqEnum[AlgChtrElt], MonStgElt ->
Example Chtr_CharacterTable (H100E1)
Example Chtr_CharacterTable2 (H100E2)
Character Ring Operations
Related Structures
NumberOfClasses(R) : AlgChtr -> RngIntElt
ClassesData(R) : AlgChtr -> SeqEnum[Tup]
Group(R) : AlgChtr -> Grp
PowerMap(R) : AlgChtr -> Map
Kernel(x) : AlgChtrElt -> Grp
Centre(x) : AlgChtrElt -> Grp
CoefficientField(x) : AlgChtrElt -> Rng
CharacterField(x) : AlgChtrElt -> Rng
DegreeOfCharacterField(x) : AlgChtrElt -> RngIntElt
Element Operations
Arithmetic
Predicates and Booleans
x in y : AlgChtrElt, AlgChtrElt -> BoolElt
x notin y : AlgChtrElt, AlgChtrElt -> BoolElt
IsCharacter(x) : AlgChtrElt -> BoolElt
IsGeneralizedCharacter(x) : AlgChtrElt -> BoolElt
IsIrreducible(x) : AlgChtrElt -> BoolElt
IsLinear(x) : AlgChtrElt -> BoolElt
IsFaithful(x) : AlgChtrElt -> BoolElt
IsReal(x) : AlgChtrElt -> BoolElt
IsPermutationCharacter(c) : AlgChtrElt -> BoolElt
Example Chtr_grp-ispermutationcharacter (H100E3)
Accessing Class Functions
T[i] : TabChtr, RngIntElt -> AlgChtrElt
T[i][j] : TabChtr, RngIntElt, RngIntElt -> FldCycElt
# T : SeqEnum -> RngIntElt
x(g) : AlgChtrElt, GrpElt -> FldCycElt
x[i] : AlgChtrElt, RngIntElt -> FldCycElt
# x : AlgChtrElt -> RngIntElt
Conjugation of Class Functions
x ^ g : AlgChtrElt, GrpElt -> AlgChtrElt
x ^ H : AlgChtrElt, Grp -> { AlgChtrElt }
GaloisConjugate(x, j) : AlgChtrElt, RngIntElt -> AlgChtrElt
GaloisOrbit(x) : AlgChtrElt -> { AlgChtrElt }
IsConjugate(x, y) : AlgChtrElt, AlgChtrElt -> BoolElt, RngIntElt
ClassPowerCharacter(x, j) : AlgChtrElt, RngIntElt -> AlgChtrElt
Functions Returning a Scalar
Degree(x) : AlgChtrElt -> RngIntElt
InnerProduct(x, y) : AlgChtrElt, AlgChtrElt -> FldCycElt
Order(x) : AlgChtrElt -> RngIntElt
Norm(x) : AlgChtrElt -> FldCycElt
Schur(x, k) : AlgChtrElt, RngIntElt -> FldCycElt
StructureConstant(G, i, j, k) : Grp, RngIntElt, RngIntElt, RngIntElt -> RngIntElt
The Schur Index
SchurIndex(x) : AlgChtrElt -> RngIntElt
SchurIndices(x) : AlgChtrElt -> SeqEnum
Example Chtr_SchurIndex (H100E4)
Example Chtr_recipe-for-schur-index (H100E5)
SchurIndexGroup(n: parameters) : RngIntElt -> GrpPC
CharacterWithSchurIndex(n: parameters) : RngIntElt -> AlgChtrElt. GrpPC
Attribute
AssertAttribute(x, "IsCharacter", b) : AlgChtrElt, MonStgElt, BoolElt ->
Constructions for Characters
Induction, Restriction and Lifting
Induction(x, G) : AlgChtrElt, Grp -> AlgChtrElt
LiftCharacter(c, f, G) : AlgChtrElt, Map, Grp -> AlgChtrElt
LiftCharacters(T, f, G) : [AlgChtrElt], Map, Grp -> AlgChtrElt
Restriction(x, H) : AlgChtrElt, Grp -> AlgChtrElt
SubgroupFusion(G, H) : Grp, Grp -> SeqEnum[RngIntElt]
Symmetrization
Symmetrization(chi, p) : AlgChtrElt, [ RngIntElt ] -> AlgChtrElt
OrthogonalSymmetrization(chi, p) : AlgChtrElt, [ RngIntElt ] -> AlgChtrElt
SymplecticSymmetrization(chi, p) : AlgChtrElt, [ RngIntElt ] -> AlgChtrElt
Example Chtr_tensprod-check (H100E6)
Example Chtr_symp-and-orthog (H100E7)
IsOrthogonalCharacter(chi) : AlgChtrElt -> BoolElt
Permutation Character
PermutationCharacter(G) : GrpPerm -> AlgChtrElt
PermutationCharacter(G, H) : Grp, Grp -> AlgChtrElt
Composition and Decomposition
Composition(T, q) : [ AlgChtrElt ], [RngElt] -> AlgChtrElt
Decomposition(T, y) : [AlgChtrElt], AlgChtrElt -> [ FldCycElt ], AlgChtrElt
Finding Irreducibles
RemoveIrreducibles(I, C) : [ AlgChtrElt ], [ AlgChtrElt ] -> [ AlgChtrElt ], [ AlgChtrElt ]
ReduceCharacters(I, C) : [ AlgChtrElt ], [ AlgChtrElt ] -> [ AlgChtrElt ], [ AlgChtrElt ]
Example Chtr_A5 (H100E8)
Brauer Characters
Structure Creation
CharacterRing(G, p) : Grp, RngIntElt -> AlgChtr
CharacterRing(R, p) : AlgChtr, RngIntElt -> AlgChtr
BrauerCharacteristic(R) : AlgChtr -> RngIntElt
Element Creation
R ! [ a1, ..., ak ] : AlgChtr, SeqEnum -> AlgChtrElt
Id(R) : AlgChtr -> AlgChtrElt
Zero(R) : AlgChtr -> AlgChtrElt
BrauerCharacter(x, p) : AlgChtrElt, RngIntElt -> AlgChtrElt
BrauerCharacter(M) : ModGrp -> AlgChtrElt
BrauerCharacterTable(G, p) : Chtr -> SeqEnum
Arithmetic
Induction, Restriction and Lifting
Induction(x, G) : AlgChtrElt, Grp -> AlgChtrElt
LiftCharacter(c, f, G) : AlgChtrElt, Map, Grp -> AlgChtrElt
LiftCharacters(T, f, G) : [AlgChtrElt], Map, Grp -> AlgChtrElt
Restriction(x, H) : AlgChtrElt, Grp -> AlgChtrElt
Blocks(T, p) : SeqEnum[AlgChtrElt], RngIntElt -> SeqEnum, SeqEnum
DefectGroup(T, b, p) : SeqEnum[AlgChtrElt], SetEnum[RngIntElt], RngIntElt -> Grp
DefectGroup(x, p) : AlgChtrElt, RngIntElt -> Grp
Example Chtr_brauer (H100E9)
Database of Character Tables
CharacterTable(N) : MonStgElt -> SeqEnum[AlgChtrElt]
HasCharacterTable(A) : GrpAtlas -> BoolElt
CharacterTableNames() : -> SetIndx[MonStElt]
Example Chtr_CharacterTableData (H100E10)
Example Chtr_Lyons-schur-index (H100E11)
Bibliography
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