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MODULES OVER AN ALGEBRA AND GROUP REPRESENTATIONS
Acknowledgements
Introduction
Constructions for A-Modules
Constructions for K[G]-Modules
General K[G]-Modules
Natural K[G]-Modules
Permutation Modules
Action on an Elementary Abelian Section
Action on a Polynomial Ring
New Modules from Old
Direct Sums and Tensor Products
Induction, Restriction and Inflation for K[G]-Modules
The Fixed-point Spaces for a K[G]-Module
Change Ring and Base Change
Writing a K[G]-Module over a Smaller Field
Rewriting Over a Smaller Degree Finite Field
Rewriting Over a Smaller Degree Number Field
Accessing Module Information
The Underlying Vector Space
The Action Algebra
Group Representations
Module Elements
Construction
Deconstruction of Module Elements
Action of the Algebra on the Module
Arithmetic with Module Elements
Indexing
Properties of Module Elements
Submodules and Quotient Modules
Construction
Membership and Equality
Operations on Submodules
Quotient Modules
Properties of a Module
Structure of a Module
Splitting a Module
Composition Series
Minimal and Maximal Submodules
Socle Series
Decomposition and Complements
Characters and Character Tables
Ordinary Characters
Brauer Characters
Constructing All Irreducible K[G]-Modules
Constructing Irreducible K[G]-Modules (Characteristic p)
Constructing Irreducible K[G]-Modules (Characteristic Zero)
The Cartan Matrix
Lattice of Submodules
Creating Lattices
Operations on Lattices
Operations on Lattice Elements
Properties of Lattice Elements
Homomorphisms
Creating Homomorphisms and Hom Spaces
Isomorphism and Similarity
Isomorphism
Similarity of Cyclic Algebras and their Modules
The Endomorphism Ring
Projective Indecomposable Modules
Cohomology and Extensions
Cohomology
Extensions of Modules
Vertex and Source of an Indecomposable Module
Bimodules
Introduction
Construction
Extracting the Left and Right Modules From a Bimodule
Permutation Bimodules
Tensor Products of Bimodules
Enumerating All Irreducible Modules
Irreducible Modules over Fq for Arbitrary Groups
Irreducible Modules over Fq for Soluble Groups
Irreducible Modules over Q for Arbitrary Groups
Modules over a General Algebra
Introduction
Construction of Algebra Modules
The Action of an Algebra Element
Related Structures of an Algebra Module
Properties of an Algebra Module
Creation of Algebra Modules from other Algebra Modules
Bibliography
Introduction
Constructions for A-Modules
RModule(A) : AlgMat -> ModRng
RModule(Q) : [ MtrxS ] -> ModTupRng
Example ModAlg_CreateK6 (H99E1)
Constructions for 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 ModAlg_CreateL27 (H99E2)
Example ModAlg_CreateMatrices (H99E3)
Natural K[G]-Modules
GModule(G, K) : GrpPerm, Rng -> ModGrp
GModule(G) : GrpMat -> ModGrp
Example ModAlg_CreateM11 (H99E4)
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 ModAlg_CreateM12 (H99E5)
Example ModAlg_CreateA7 (H99E6)
Action on an Elementary Abelian Section
GModule(G, A, B) : Grp, Grp, Grp -> ModGrp, Map
Example ModAlg_CreateA4wrC3 (H99E7)
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 ModAlg_CreatePolyAction (H99E8)
New Modules from Old
Direct Sums and Tensor Products
DirectSum(M, N) : ModRng, ModRng -> ModRng, Map, Map, Map, Map
DirectSum(Q) : [ ModRng ] -> ModRng, [ Map ], [ Map ]
TensorProduct(M, N) : ModMat, ModMat -> ModMat
TensorProduct(M, N) : ModGrp, ModGrp -> ModGrp
TensorPower(M, n) : ModMat, RngIntElt -> ModMat
ExteriorSquare(M) : ModMat -> ModMat
SymmetricSquare(M) : ModMat -> ModMat
GTensorProduct(M, N) : ModGrp, ModGrp -> ModGrp, Map
GTensorProduct(M, N, H) : ModGrp, ModGrp, Grp -> ModGrp, Map
Induction, Restriction and Inflation for K[G]-Modules
Dual(M) : ModGrp -> ModGrp
Induction(M, G) : ModGrp, Grp -> ModGrp
Induction(R, G) : Map, Grp -> Map
Restriction(M, H) : ModGrp, Grp -> ModGrp
Inflation(M, h) : ModGrp, Map -> ModGrp
Example ModAlg_GModules1 (H99E9)
The Fixed-point Spaces for a K[G]-Module
Fix(M): Mod -> Mod
FixMod(M, H): ModGrp, Grp -> Mod
FixDual(M): ModGrp -> Modgrp, Map
FixDualMod(M,H): ModGrp, Grp -> Mod
Change Ring and Base Change
ChangeRing(M, S) : ModRng, Rng -> ModRng, Map
ChangeRing(M, S, f) : ModRng, Rng, Map -> ModRng, Map
M ^ T : ModGrp, AlgMatElt -> ModGrp
Writing a K[G]-Module over a Smaller Field
Rewriting Over a Smaller Degree Finite 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
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
Rewriting Over a Smaller Degree Number Field
AbsoluteModuleOverMinimalField(M) : ModGrp -> ModGrp
Minimize(M) : ModGrp -> ModGrp
WriteGModuleOver(M, K) : ModGrp, FldAlg -> ModGrp
Example ModAlg_minimal-field (H99E10)
Accessing Module Information
The Underlying Vector Space
M . i : ModRng, RngIntElt -> ModElt
CoefficientRing(M) : ModRng -> Rng
Generators(M) : ModRng -> { ModRngElt }
Parent(u) : ModRngElt -> ModRng
The Action Algebra
Action(M) : ModRng -> AlgMat
MatrixGroup(M) : ModGrp -> GrpMat
ActionGenerator(M, i) : ModRng, RngIntElt -> AlgMatElt
NumberOfActionGenerators(M) : ModRng -> RngIntElt
Group(M) : ModGrp -> Grp
Example ModAlg_Access (H99E11)
Group Representations
GModuleAction(M) : ModGrp -> Map(Hom)
Representation(M) : ModGrp -> Map(Hom)
Example ModAlg_Representation (H99E12)
Example ModAlg_Dual (H99E13)
Kernel(M) : ModGrp -> Grp
GModuleOfQuotient(M, H) : ModGrp, Grp -> ModGrp
ActionGenerator(M, i) : ModGrp, RngIntElt -> AlgMatElt
ActionGenerators(M) : ModGrp -> [ AlgMatElt ]
NumberOfActionGenerators(M) : ModGrp -> RngIntElt
ActionGroup(M) : ModGrp -> GrpMat
Sections(G) : GrpMat -> List
Example ModAlg_Sections (H99E14)
Module Elements
Construction
elt< M | a1, ..., an > : ModRng, List -> ModRngElt
M ! Q : ModRng, [RngElt] -> ModRngElt
Zero(M) : ModRng -> ModRngElt
Random(M) : ModRng -> ModRngElt
Deconstruction of Module Elements
ElementToSequence(u) : ModRngElt -> [RngElt]
Action of the Algebra on the Module
u * a : ModRngElt, AlgElt -> ModRngElt
u * g : ModGrpElt, GrpElt -> ModGrpElt
Arithmetic with Module Elements
u + v : ModRngElt, ModRngElt -> ModRngElt
- u : ModRngElt -> ModRngElt
u - v : ModRngElt, ModRngElt -> ModRngElt
k * u : RngElt, ModRngElt -> ModRngElt
u * k : ModRngElt, RngElt -> ModRngElt
u / k : ModRngElt, RngElt -> ModRngElt
Indexing
u[i] : ModRngElt, RngIntElt -> RngElt
u[i] := x : ModRngElt, RngIntElt, RngElt -> ModRngElt
Properties of Module Elements
IsZero(u) : ModRngElt -> BoolElt
Support(u) : ModRngElt -> { RngIntElt }
Submodules and Quotient Modules
Construction
sub<M | L> : ModRng, List -> ModRng
ImageWithBasis(X, M) : ModMatRngElt, ModRng -> ModRng
Morphism(M, N) : ModRng, ModRng -> ModMatRngElt
Example ModAlg_Submodule (H99E15)
Membership and Equality
u in M : ModRngElt, ModRng -> BoolElt
N subset M : ModRng, ModRng -> BoolElt
N eq M : ModRng, ModRng -> BoolElt
Operations on Submodules
M + N : ModRng, ModRng -> ModRng
M meet N : ModRng, ModRng -> ModRng
Quotient Modules
quo<M | L> : ModRng, List -> ModRng
Morphism(M, N) : ModRng, ModRng -> ModMatRngElt
Example ModAlg_QuotientModule (H99E16)
Properties of a Module
IsIrreducible(M) : ModRng -> BoolElt, ModRng, ModRng
IsAbsolutelyIrreducible(M) : ModRng -> BoolElt, AlgMatElt, RngIntElt
IsDecomposable(M) : ModRng -> BoolElt, ModRng, ModRng
IsSemisimple(M) : ModGrp -> BoolElt
IsProjective(M) : ModGrp -> BoolElt
IsFree(M) : ModGrp -> BoolElt
IsSelfDual(M) : ModGrp -> BoolElt
IsPermutationModule(M) : ModRng -> BoolElt
Structure of a Module
Splitting a Module
Meataxe(M) : ModRng -> ModRng, ModRng, AlgMatElt
IsIrreducible(M) : ModRng -> BoolElt, ModRng, ModRng
IsAbsolutelyIrreducible(M) : ModRng -> BoolElt, AlgMatElt, RngIntElt
AbsolutelyIrreducibleModule(M) : ModRng -> ModRng
MinimalField(M) : ModRng -> FldFin
Example ModAlg_Meataxe (H99E17)
Composition Series
CompositionSeries(M) : ModRng -> [ ModRng ], [ ModRng ], AlgMatElt
CompositionFactors(M) : ModRng -> [ ModRng ]
Constituents(M) : ModRng -> [ ModRng ], [ RngIntElt ]
ConstituentsWithMultiplicities(M) : ModRng -> [ <ModRng, RngIntElt> ], [ RngIntElt ]
Example ModAlg_CompSeries (H99E18)
Minimal and Maximal Submodules
JacobsonRadical(M) : ModRng -> ModRng, Map
MinimalSubmodule(M) : ModRng -> ModRng
MinimalSubmodules(M) : ModRng -> [ ModRng ], BoolElt
MinimalSubmodules(M, F) : ModRng, ModRng -> [ ModRng ], BoolElt
MaximalSubmodules(M) : ModRng -> [ ModRng ], BoolElt
Socle Series
Socle(M) : ModRng -> ModRng, Map
SocleRecursive(M) : ModRng -> ModRng
SocleSeries(M) : ModRng -> [ ModRng ], [ ModRng ], AlgMatElt
SocleFactors(M) : ModRng -> [ ModRng ]
SocleLayerFactors(M) : ModGrp -> SeqEnum
DisplaySocleStructure(SL) : SeqEnum ->
Example ModAlg_Minimals (H99E19)
Decomposition and Complements
IsDecomposable(M) : ModRng -> BoolElt, ModRng, ModRng
IsSemisimple(M) : ModGrp -> BoolElt
DirectSumDecomposition(M) : ModRng -> [ ModRng ]
RelativeDecomposition(M, T) : ModRng, ModRng) -> ModRng, ModRng
HasComplement(M, S) : ModGrp, ModGrp -> BoolElt, ModGrp
Complements(M, S) : ModGrp, ModGrp -> [ ModGrp ]
Example ModAlg_Decomposable (H99E20)
Characters and Character Tables
Ordinary Characters
Character(M) : ModGrp -> AlgChtrElt
CharacterTable(G) : Chtr -> SeqEnum
SymmetricCharacterTable(n) : RngIntElt -> SeqEnum
RationalCharacterTable(G) : Chtr -> SeqEnum
Brauer Characters
BrauerCharacterTable(G, p) : Chtr -> SeqEnum
Character(M) : ModGrp -> AlgChtrElt
Constructing All Irreducible K[G]-Modules
Constructing Irreducible K[G]-Modules (Characteristic p)
IrreducibleModules(G, K : parameters) : Grp, Fld -> SeqEnum
AbsolutelyIrreducibleModules(G, K : parameters) : Grp, Fld -> SeqEnum
AbsolutelyIrreducibleModulesSoluble(G, p) : Grp, RngIntElt -> SeqEnum
AbsolutelyIrreducibleModulesSchur(G, K) : GrpPC, Fld -> List
Example ModAlg_Irreducibles (H99E21)
Constructing Irreducible K[G]-Modules (Characteristic Zero)
GModule(x) : Chtr -> ModGrp
IrreducibleModules(G, K) : Grp, Fld -> SeqEnum
AbsolutelyIrreducibleModules(G) : Grp -> SeqEnum
AbsolutelyIrreducibleModulesSchur(G, K) : GrpPC, Fld -> List
Example ModAlg_Irreducibles (H99E22)
The Cartan Matrix
CartanMatrix(G, K) : Grp, FldFin -> AlgMatElt
AbsoluteCartanMatrix(G, K) : Grp, FldFin -> AlgMatElt
DecompositionMatrix(G, K) : Grp, FldFin -> AlgMatElt
Lattice of Submodules
Creating Lattices
SubmoduleLattice(M) : ModRng -> SubModLat, BoolElt
SubmoduleLatticeAbort(M, n) : ModRng, RngIntElt -> BoolElt, SubModLat
SetVerbose("SubmoduleLattice", i) : MonStgElt, RngIntElt ->
Submodules(M) : ModRng -> [ModRng]
Example ModAlg_CreateLattice (H99E23)
Operations on Lattices
# L : SubModLat -> RngIntElt
L ! i: SubModLat, RngIntElt -> SubModLatElt
L ! S: SubModLat, ModRng -> SubModLatElt
Bottom(L): SubModLat -> SubModLatElt
Random(L): SubModLat -> SubModLatElt
Top(L): SubModLat -> SubModLatElt
Operations on Lattice Elements
IntegerRing() ! e : RngInt, SubModLatElt -> RngIntElt
e + f : SubModLatElt, SubModLatElt -> SubModLatElt
e meet f : SubModLatElt, SubModLatElt -> SubModLatElt
e eq f : SubModLatElt, SubModLatElt -> SubModLatElt
e subset f : SubModLatElt, SubModLatElt -> SubModLatElt
MaximalSubmodules(e) : SubModLatElt -> { SubModLatElt }
MinimalSupermodules(e) : SubModLatElt -> { SubModLatElt }
Module(e) : SubModLatElt -> ModRng
Properties of Lattice Elements
Dimension(e) : SubModLatElt -> RngIntElt
JacobsonRadical(e) : SubModLatElt -> SubModLatElt
Morphism(e) : SubModLatElt -> ModMatRngElt
Example ModAlg_LatticeOps (H99E24)
Homomorphisms
Creating Homomorphisms and Hom Spaces
hom< M -> N | X > : ModRng, ModRng, ModMatElt -> Map
Hom(M, N) : ModRng, ModRng -> ModMatRng
GHom(M, N) : ModGrp, ModGrp -> ModMatGrp
GHomOverCentralizingField(M, N) : ModGrp, ModGrp -> ModMatGrp
AHom(M, N) : ModRng, ModRng -> ModMatRng
HomMod(M, N) : ModGrp, ModGrp -> ModGrp
H ! f : ModMatRng, Map -> ModMatRngElt
IsModuleHomomorphism(X) : ModMatFldElt -> BoolElt
Example ModAlg_EndoRing (H99E25)
Example ModAlg_CreateHomGHom (H99E26)
Isomorphism and Similarity
Isomorphism
IsIsomorphic(M, N) : ModRng, ModRng -> ModRng, ModRng, BoolElt, AlgMatElt
SummandIsomorphism(M, N) : ModRng, ModRng -> ModRng, ModRng, Map, Map
Similarity of Cyclic Algebras and their Modules
IsCyclic(R) : AlgAss -> BoolElt, AlgAssElt
IsSimilar(M, N) : ModRng, ModRng -> BoolElt, Map
Example ModAlg_Star_Alg (H99E27)
The Endomorphism Ring
EndomorphismAlgebra(M) : ModRng -> AlgMat
CentreOfEndomorphismRing(M) : ModRng -> AlgMat
AutomorphismGroup(M) : ModRng -> GrpMat
Example ModAlg_EndoRing (H99E28)
Projective Indecomposable Modules
ProjectiveIndecomposableDimensions(G, K) : Grp, FldFin -> SeqEnum
ProjectiveIndecomposableModule(I: parameters) : ModGrp -> ModGrp
ProjectiveIndecomposableModules(G, K: parameters) : Grp, FldFin -> SeqEnum
PIMBlocks(CM) : AlgMatElt -> SeqEnum
ProjectiveCover(M) : ModGrp -> ModGrp, ModMatGrpElt
InjectiveHull(M) : ModGrp -> ModGrp, ModMatGrpElt
Example ModAlg_Projective Indecomposables (H99E29)
Cohomology and Extensions
Cohomology
CohomologyModule(G, M) : GrpPerm, ModGrp -> ModCoho
CohomologyGroup(CM, n) : ModCoho, RngIntElt -> ModTupRng
CohomologicalDimension(CM, n) : ModCoho, RngIntElt -> RngIntElt
CohomologicalDimension(M, n) : ModGrp, n -> RngIntElt
CohomologicalDimensions(M, n) : ModGrp, n -> RngIntElt
Example ModAlg_Cohomology Group (H99E30)
Example ModAlg_Cohomological Dimension (H99E31)
Extensions of Modules
Ext(M, N) : ModGrp, ModGrp -> ModTupFld
Extension(M, N, e, r) : ModGrp, ModGrp, ModTupFldElt, Map -> ModGrp, ModMatGrpElt, ModMatGrpElt
MaximalExtension(M, N, E, r) : ModGrp, ModGrp, ModTupFld, map -> ModGrp
MaximalExtension(M, N) : ModGrp, ModGrp -> ModGrp
MaximalExtension(~M, N) : ModGrp, ModGrp ->
Example ModAlg_ModuleExtensions (H99E32)
LowDimensionalModules(G, K, n) : Grp, Fld, RngIntElt -> SeqEnum
Vertex and Source of an Indecomposable Module
Vertex(M : parameters) : ModGrp -> Grp
Source(M : parameters) : ModGrp -> ModGrp, ModGrp
Example ModAlg_Vertices and Sources in $M_{22 (H99E33)
Bimodules
Introduction
Construction
Bimodule(M,N) : ModGrp, ModGrp -> LRModGrp
Example ModAlg_small-bimodule (H99E34)
Bimodule(G,H,M) : Grp, Grp, ModGrp -> LRModGrp
Example ModAlg_dp-bimodule (H99E35)
LeftBimodule(M): ModGrp -> LRModGrp
RightBimodule(M): ModGrp -> LRModGrp
Extracting the Left and Right Modules From a Bimodule
LeftOppositeModule(B): LRModGrp -> ModGrp
RightModule(B) LRModGrp: -> ModGrp
Example ModAlg_dp-bimodule-cont (H99E36)
Permutation Bimodules
PermutationBimodule(G,H,m,K): Grp, Grp, Map, Fld -> LRModGrp
Example ModAlg_permbimodule (H99E37)
Tensor Products of Bimodules
TensorProduct(B1, B2): LRModGrp, LRModGrp -> LRModGrp
Example ModAlg_induction (H99E38)
Enumerating All Irreducible Modules
Irreducible Modules over Fq for Arbitrary Groups
IrreducibleModules(G, K : parameters) : Grp, Fld -> SeqEnum
Example ModAlg_IrreducibleModules (H99E39)
AbsolutelyIrreducibleModulesBurnside(G, K : parameters ) : Grp, FldFin -> [ ModGrp ]
IrreducibleModulesBurnside(G, K : parameters ) : Grp, FldFin -> [ ModGrp ]
AbsolutelyIrreducibleConstituents(M) : ModGrp -> [ ModGrp ]
Example ModAlg_IrreducibleModules_M11 (H99E40)
Irreducible Modules over Fq for Soluble Groups
IrreducibleModules(G, K : parameters) : Grp, Fld -> SeqEnum
IrreducibleModulesSchur(G, K: parameters) : GrpPC, Rng -> List[GModule]
Example ModAlg_Reps (H99E41)
Irreducible Modules over Q for Arbitrary Groups
IrreducibleModules(G, Q : parameters) : Grp, FldRat -> SeqEnum, SeqEnum
RationalCharacterTable(G) : Grp -> SeqEnum, SeqEnum
Example ModAlg_IrreducibleModules (H99E42)
Example ModAlg_IrreducibleModules2 (H99E43)
Modules over a General Algebra
Introduction
Construction of Algebra Modules
Module(A, m): Alg, Map[SetCart, ModRng] -> ModAlg
Example ModAlg_AlgModCreate (H99E44)
The Action of an Algebra Element
a ^ v : AlgElt, ModAlgElt -> ModAlgElt
v ^ a : ModAlgElt, AlgElt -> ModAlgElt
ActionMatrix(M, a): ModAlg, AlgElt -> AlgMatElt
Example ModAlg_Action (H99E45)
Related Structures of an Algebra Module
Algebra(M): ModAlg -> Alg
CoefficientRing(M): ModAlg -> Fld
Basis(M): ModAlg -> SeqEnum
Properties of an Algebra Module
IsLeftModule(M): ModAlg -> BoolElt
IsRightModule(M): ModAlg -> BoolElt
Dimension(M): ModAlg -> RngIntElt
Creation of Algebra Modules from other Algebra Modules
DirectSum(Q): SeqEnum -> ModAlg, SeqEnum, SeqEnum
SubalgebraModule(B, M): Alg, ModAlg -> ModAlg
ModuleWithBasis(Q): SeqEnum -> ModAlg
Example ModAlg_OtherMod (H99E46)
sub< M | S > : ModAlg, [ModAlgElt] -> ModAlg
quo< M | S > : ModAlg, [ModAlgElt] -> ModAlg
Bibliography
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