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K[G]-MODULES AND GROUP REPRESENTATIONS

 
Acknowledgements
 
Introduction
 
Construction of K[G]-Modules
      General K[G]-Modules
      Natural K[G]-Modules
      Action on an Elementary Abelian Section
      Permutation Modules
      Action on a Polynomial Ring
 
The Representation Afforded by a K[G]-module
 
Standard Constructions
      Changing the Coefficient Ring
      Writing a Module over a Smaller Field
      Direct Sum
      Tensor Products of K[G]-Modules
      Induction and Restriction
      The Fixed-point Space of a Module
      Changing Basis
 
The Construction of all Irreducible Modules
      Generic Functions for Finding Irreducible Modules
      The Burnside Algorithm
      The Schur Algorithm for Soluble Groups
      The Rational Algorithm
 
Extensions of Modules
 
The Construction of Projective Indecomposable Modules







DETAILS

 
Introduction

 
Construction of K[G]-Modules

      General K[G]-Modules
            GModule(G, A) : Grp, AlgMat -> ModGrp
            GModule(G, Q) : Grp, [ GrpMatElt ] -> ModGrp
            TrivialModule(G, K) : Grp, Fld -> ModGrp
            Example ModGrp_CreateL27 (H90E1)
            Example ModGrp_CreateMatrices (H90E2)

      Natural K[G]-Modules
            GModule(G, K) : GrpPerm, Rng -> ModGrp
            GModule(G) : GrpMat -> ModGrp
            Example ModGrp_CreateM11 (H90E3)

      Action on an Elementary Abelian Section
            GModule(G, A, B) : Grp, Grp, Grp -> ModGrp, Map
            Example ModGrp_CreateA4wrC3 (H90E4)

      Permutation Modules
            PermutationModule(G, H, K) : Grp, Grp, Fld -> ModGrp
            PermutationModule(G, K) : Grp, Fld -> ModGrp
            PermutationModule(G, V) : Grp, ModTupFld -> ModGrp
            PermutationModule(G, u) : Grp, ModTupFldElt -> ModGrp
            Example ModGrp_CreateM12 (H90E5)
            Example ModGrp_CreateA7 (H90E6)

      Action on a Polynomial Ring
            GModule(G, P, d) : Grp, RngMPol, RngIntElt -> ModGrp, Map, @ RngMPolElt @
            GModule(G, I, J) : Grp, RngMPol, RngMPol -> ModGrp, Map, @ RngMPolElt @
            GModule(G, Q) : Grp, RngMPolRes -> ModGrp, Map, @ RngMPolElt @
            Example ModGrp_CreatePolyAction (H90E7)

 
The Representation Afforded by a K[G]-module
      GModuleAction(M) : ModGrp -> Map(Hom)
      Representation(M) : ModGrp -> Map(Hom)
      Example ModGrp_Representation (H90E8)
      Example ModGrp_Dual (H90E9)
      ActionGenerator(M, i) : ModGrp, RngIntElt -> AlgMatElt
      ActionGenerators(M) : ModGrp -> [ AlgMatElt ]
      NumberOfActionGenerators(M) : ModGrp -> RngIntElt
      ActionGroup(M) : ModGrp -> GrpMat
      Sections (G) : GrpMat -> List
      Example ModGrp_Sections (H90E10)

 
Standard Constructions

      Changing the Coefficient Ring
            ChangeRing(M, S) : ModRng, Rng -> ModRng, Map
            ChangeRing(M, S, f) : ModRng, Rng, Map -> ModRng, Map

      Writing a Module over a Smaller Field
            IsRealisableOverSmallerField(M) : ModGrp -> BoolElt, ModGrp
            IsRealisableOverSubfield(M, F) : ModGrp, FldFin -> BoolElt, ModGrp
            WriteOverSmallerField(M, F) : ModGrp, FldFin -> ModGrp, Map
            AbsoluteModuleOverMinimalField(M, F) : ModGrp, FldFin -> ModGrp
            AbsoluteModuleOverMinimalField(M) : ModGrp -> ModGrp
            Minimize(R) : Map -> Map
            AbsoluteModulesOverMinimalField(Q, F) : [ ModGrp ], FldFin -> [ ModGrp ]
            ModuleOverSmallerField(M, F) : ModGrp, FldFin -> ModGrp
            ModulesOverSmallerField(Q, F) : SeqEnum, FldFin -> ModGrp
            ModulesOverCommonField(M, N) : ModGrp, ModGrp -> ModGrp, ModGrp
            WriteGModuleOver(M, K) : ModGrp, FldAlg -> ModGrp
            WriteRepresentationOver(R, K) : Map, FldAlg -> Map
            Example ModGrp_minimal-field (H90E11)

      Direct Sum
            DirectSum(M, N) : ModGrp, ModGrp -> ModGrp, Map, Map, Map, Map
            DirectSum(Q) : [ ModGrp ] -> [ ModGrp ], [ Map ], [ Map ]

      Tensor Products of K[G]-Modules
            TensorProduct(M, N) : ModGrp, ModGrp -> ModGrp
            TensorPower(M, n) : ModGrp, RngIntElt -> ModGrp
            ExteriorSquare(M) : ModGrp -> ModGrp
            SymmetricSquare(M) : ModGrp -> ModGrp

      Induction and Restriction
            Dual(M) : ModGrp -> ModGrp
            Induction(M, G) : ModGrp, Grp -> ModGrp
            Induction(R, G) : Map, Grp -> Map
            Restriction(M, H) : ModGrp, Grp -> ModGrp
            Example ModGrp_GModules1 (H90E12)

      The Fixed-point Space of a Module
            Fix(M): Mod -> Mod

      Changing Basis
            M ^ T : ModGrp, AlgMatElt -> ModGrp

 
The Construction of all Irreducible Modules

      Generic Functions for Finding Irreducible Modules
            IrreducibleModules(G, K : parameters) : Grp, Fld -> SeqEnum
            Example ModGrp_IrreducibleModules (H90E13)

      The Burnside Algorithm
            AbsolutelyIrreducibleModulesBurnside(G, K : parameters ) : Grp, FldFin -> [ ModGrp ]
            IrreducibleModulesBurnside(G, K : parameters ) : Grp, FldFin -> [ ModGrp ]
            AbsolutelyIrreducibleConstituents(M) : ModGrp -> [ ModGrp ]
            Example ModGrp_IrreducibleModules_M11 (H90E14)

      The Schur Algorithm for Soluble Groups
            IrreducibleModules(G, K : parameters) : Grp, Fld -> SeqEnum
            IrreducibleModulesSchur(G, K: parameters) : GrpPC, Rng -> List[GModule]
            Example ModGrp_Reps (H90E15)
            AbsolutelyIrreducibleRepresentationsInit(G, F : parameters) : GrpPC, Fld -> SolRepProc
            NextRepresentation(P) : SolRepProc -> BoolElt, Map
            AbsolutelyIrreducibleRepresentationProcessDelete(~P) : SolRepProc ->

      The Rational Algorithm
            IrreducibleModules(G, Q : parameters) : Grp, FldRat -> SeqEnum, SeqEnum
            RationalCharacterTable(G) : Grp -> SeqEnum, SeqEnum
            Example ModGrp_IrreducibleModules (H90E16)
            Example ModGrp_IrreducibleModules2 (H90E17)

 
Extensions of Modules
      Ext(M, N) : ModGrp, ModGrp -> ModTupFld
      Extension(M, N, e) : ModGrp, ModGrp, ModTupFldElt -> ModGrp, ModMatGrpElt, ModMatGrpElt
      MaximalExtension(M, N, E) : ModGrp, ModGrp, ModTupFld -> ModGrp
      Example ModGrp_ModuleExtensions (H90E18)

 
The Construction of Projective Indecomposable Modules
      ProjectiveIndecomposableDimensions(G, K) : Grp, FldFin -> SeqEnum
      ProjectiveIndecomposableModule(I: parameters) : ModGrp -> ModGrp
      ProjectiveIndecomposableModules(G, K: parameters) : Grp, FldFin -> SeqEnum
      Example ModGrp_Projective Indecomposables (H90E19)
      CartanMatrix(G, K) : Grp, FldFin -> AlgMatElt
      AbsoluteCartanMatrix(G, K) : Grp, FldFin -> AlgMatElt
      DecompositionMatrix(G, K) : Grp, FldFin -> AlgMatElt
      ProjectiveCover(M) : ModGrp -> ModGrp, ModMatGrpElt
      CohomologicalDimension(M, n) : ModGrp, n -> RngIntElt
      CohomologicalDimensions(M, n) : ModGrp, n -> RngIntElt
      Example ModGrp_Cohomological Dimension (H90E20)

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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013