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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
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 (H89E1)
Example ModGrp_CreateMatrices (H89E2)
Natural K[G]-Modules
GModule(G, K) : GrpPerm, Rng -> ModGrp
GModule(G) : GrpMat -> ModGrp
Example ModGrp_CreateM11 (H89E3)
Action on an Elementary Abelian Section
GModule(G, A, B) : Grp, Grp, Grp -> ModGrp, Map
Example ModGrp_CreateA4wrC3 (H89E4)
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 (H89E5)
Example ModGrp_CreateA7 (H89E6)
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 (H89E7)
The Representation Afforded by a K[G]-module
GModuleAction(M) : ModGrp -> Map(Hom)
Representation(M) : ModGrp -> Map(Hom)
Example ModGrp_Representation (H89E8)
Example ModGrp_Dual (H89E9)
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 (H89E10)
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 (H89E11)
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 (H89E12)
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 (H89E13)
The Burnside Algorithm
AbsolutelyIrreducibleModulesBurnside(G, K : parameters ) : Grp, FldFin -> [ ModGrp ]
IrreducibleModulesBurnside(G, K : parameters ) : Grp, FldFin -> [ ModGrp ]
AbsolutelyIrreducibleConstituents(M) : ModGrp -> [ ModGrp ]
Example ModGrp_IrreducibleModules (H89E14)
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 (H89E15)
AbsolutelyIrreducibleRepresentationsInit(G, F : parameters) : GrpPC, Fld -> SolRepProc
NextRepresentation(P) : SolRepProc -> BoolElt, Map
AbsolutelyIrreducibleRepresentationProcessDelete(~P) : SolRepProc ->
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