K[G]-MODULES AND GROUP REPRESENTATIONS

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 (H78E1)
            Example ModGrp_CreateMatrices (H78E2)

      Natural K[G]-Modules
            GModule(G, K) : GrpPerm, Rng -> ModGrp
            GModule(G) : GrpMat -> ModGrp
            Example ModGrp_CreateM11 (H78E3)

      Action on an Elementary Abelian Section
            GModule(G, A, B) : Grp, Grp, Grp -> ModGrp, Map
            Example ModGrp_CreateA4wrC3 (H78E4)

      Permutation Modules
            PermutationModule(G, H, K) : Grp, Grp, Fld -> ModGrp
            PermutationModule(G, K) : Grp, Fld -> ModGrp
            PermutationModule(G, V) : Grp, ModTup -> ModGrp
            PermutationModule(G, u) : Grp, ModTupElt -> ModGrp
            Example ModGrp_CreateM12 (H78E5)
            Example ModGrp_CreateA7 (H78E6)

      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 (H78E7)

 
The Representation Afforded by a K[G]-module
      GModuleAction(M) : ModGrp -> Map(Hom)
      Representation(M) : ModGrp -> Map(Hom)
      ActionGenerator(M, i) : ModGrp, RngIntElt -> AlgMatElt
      ActionGenerators(M) : ModGrp -> [ AlgMatElt ]
      NumberOfActionGenerators(M) : ModGrp -> RngIntElt
      Example ModGrp_Representation (H78E8)
      Example ModGrp_Dual (H78E9)

 
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
            AbsoluteModulesOverMinimalField(Q, K) : [ ModGrp ], FldFin -> [ ModGrp ]
            ModuleOverSmallerField(M, F) : ModGrp, FldFin -> ModGrp
            ModulesOverSmallerField(Q, F) : SeqEnum, FldFin -> ModGrp
            ModulesOverCommonField(M, N) : ModGrp, ModGrp -> ModGrp, ModGrp

      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 (H78E10)

      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_Extension (H78E11)

      The Burnside Algorithm for General Groups
            AbsolutelyIrreducibleModulesBurnside(G, K : parameters ) : Grp, FldFin -> [ ModGrp ]
            IrreducibleModulesBurnside(G, K : parameters ) : Grp, FldFin -> [ ModGrp ]
            AbsolutelyIrreducibleConstituents(M) : ModGrp -> [ ModGrp ]
            Example ModGrp_IrreducibleModules (H78E12)

      The Schur Algorithm for Soluble Groups
            AbsolutelyIrreducibleModulesSchur(G, K: parameters) : GrpPC, Rng -> List[GModule]
            IrreducibleModulesSchur(G, K: parameters) : GrpPC, Rng -> List[GModule]
            Example ModGrp_Reps (H78E13)
            AbsolutelyIrreducibleRepresentationsInit(G, F : parameters) : GrpPC, Fld -> SolRepProc
            NextRepresentation(P) : SolRepProc -> BoolElt, Map

 
Bibliography