MAGMA Computational Algebra System

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 are represented by the k values (elements of some cyclotomic field Q(ζ_n)) on the classes.  
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
      Induction, Restriction and Lifting
      Symmetrization
      Permutation Character
      Composition and Decomposition
      Finding Irreducibles
      Brauer Characters
 
Bibliography

DETAILS

 
Creation Functions

      Structure Creation
            ClassFunctionSpace(G) : Grp -> AlgChtr

      Element Creation
            elt< R | a₁, ..., a_k :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
            LinearCharacters(G): Grp -> SeqEnum
            CharacterDegrees(G): GrpPerm -> SeqEnum
            CharacterDegrees(G, z, p): GrpPC, GrpPCElt, RngIntElt -> SeqEnum
            CharacterDegreesPGroup(G): GrpPC -> SeqEnum
            RationalCharacterTable(G): GrpFin -> SeqEnum

 
Character Ring Operations

      Related Structures
            Group(R) : AlgChtr -> Grp
            Centre(x) : AlgChtrElt -> Grp
            CoefficientField(x) : AlgChtrElt -> Rng
            Kernel(x) : AlgChtrElt -> Grp

 
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

      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 }
            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: parameters) : AlgChtrElt -> SeqEnum
            Example Chtr_SchurIndex (H90E1)

      Attribute
            AssertAttribute(x, "IsCharacter", b) : AlgChtrElt, MonStgElt, BoolElt ->

      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

      Symmetrization
            Symmetrization(x, p) : AlgChtrElt, [ RngIntElt ] -> AlgChtrElt
            OrthogonalComponent(x, p) : AlgChtrElt, [ RngIntElt ] -> AlgChtrElt
            SymplecticComponent(x, p) : AlgChtrElt, [ RngIntElt ] -> AlgChtrElt
            SymmetricComponents(x, n) : AlgChtrElt, RngIntElt -> SetEnum
            OrthogonalComponents(x, n) : AlgChtrElt, RngIntElt -> SetEnum
            SymplecticComponents(x, n) : AlgChtrElt, RngIntElt -> SetEnum

      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 (H90E2)

      Brauer Characters
            BrauerCharacter(x, p) : AlgChtrElt, RngIntElt -> AlgChtrElt
            Blocks(T, p) : SeqEnum[AlgChtrElt], RngIntElt -> SeqEnum, SeqEnum

 
Bibliography

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