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Acknowledgements
Introduction
Modules over a Matrix Algebra
Construction of an A-Module
General Constructions
Constructions for K[G]-Modules
Accessing Module Information
The Underlying Vector Space
The Algebra
Standard Constructions
Changing the Coefficient Ring
Direct Sum
Changing Basis
Element Construction and Operations
Construction of Module Elements
Deconstruction of Module Elements
Action of the Algebra on the Module
Arithmetic with Module Elements
Indexing
Properties of Module Elements
Submodules
Construction
Membership and Equality
Operations on Submodules
Quotient Modules
Structure of a Module
Reducibility
Composition Series
Socle Series
Decomposability and Complements
Lattice of Submodules
Creating Lattices
Operations on Lattices
Operations on Lattice Elements
Properties of Lattice Elements
Homomorphisms
Creating Homomorphisms
Hom(M, N)
Endo-- and Automorphisms
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
DETAILS
Introduction
Modules over a Matrix Algebra
Construction of an A-Module
General Constructions
RModule(A) : AlgMat -> ModRng
RModule(Q) : [ MtrxS ] -> ModTupRng
Example ModAlg_CreateK6 (H86E1)
Constructions for K[G]-Modules
GModule(G, Q) : Grp, [ MtrxS ] -> ModGrp
PermutationModule(G, K) : GrpPerm, Fld -> ModGrp
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 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 (H86E2)
Standard Constructions
Changing the Coefficient Ring
ChangeRing(M, S) : ModRng, Rng -> ModRng, Map
ChangeRing(M, S, f) : ModRng, Rng, Map -> ModRng, Map
Direct Sum
DirectSum(M, N) : ModRng, ModRng -> ModRng, Map, Map, Map, Map
DirectSum(Q) : [ ModRng ] -> ModRng, [ Map ], [ Map ]
Changing Basis
M ^ T : ModRng, AlgMatElt -> ModRng
Element Construction and Operations
Construction of Module Elements
elt< M | a1, ..., an > : ModRng, List -> ModRngElt
M ! Q : ModRng, [RngElt] -> ModRngElt
Zero(M) : ModRng, RngIntElt -> 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
Construction
sub<M | L> : ModRng, List -> ModRng
ImageWithBasis(X, M) : ModMatRngElt, ModRng -> ModRng
Morphism(M, N) : ModRng, ModRng -> ModMatRngElt
Example ModAlg_Submodule (H86E3)
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 (H86E4)
Structure of a Module
Reducibility
Meataxe(M) : ModRng -> ModRng, ModRng, AlgMatElt
IsIrreducible(M) : ModRng -> BoolElt, ModRng, ModRng
IsAbsolutelyIrreducible(M) : ModRng -> BoolElt, AlgMatElt, RngIntElt
AbsolutelyIrreducibleModule(M) : ModRng -> ModRng
Example ModAlg_Meataxe (H86E5)
MinimalField(M) : ModRng -> FldFin
IsPermutationModule(M) : ModRng -> BoolElt
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 (H86E6)
Socle Series
IsSemisimple(M) : ModGrp -> BoolElt
MaximalSubmodules(M) : ModRng -> [ ModRng ], BoolElt
JacobsonRadical(M) : ModRng -> ModRng, Map
MinimalSubmodules(M) : ModRng -> [ ModRng ], BoolElt
MinimalSubmodules(M, F) : ModRng, ModRng -> [ ModRng ], BoolElt
MinimalSubmodule(M) : ModRng -> ModRng
Socle(M) : ModRng -> ModRng, Map
SocleSeries(M) : ModRng -> [ ModRng ], [ ModRng ], AlgMatElt
SocleFactors(M) : ModRng -> [ ModRng ]
Example ModAlg_Minimals (H86E7)
Decomposability and Complements
IsDecomposable(M) : ModRng -> BoolElt, ModRng, ModRng
DirectSumDecomposition(M) : ModRng -> [ ModRng ]
HasComplement(M, S) : ModGrp, ModGrp -> BoolElt, ModGrp
Complements(M, S) : ModGrp, ModGrp -> [ ModGrp ]
Example ModAlg_Decomposable (H86E8)
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 (H86E9)
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 (H86E10)
Homomorphisms
Creating Homomorphisms
hom< M -> N | X > : ModRng, ModRng, ModMatElt -> Map
H ! f : ModMatRng, Map -> ModMatRngElt
IsModuleHomomorphism(X) : ModMatFldElt -> BoolElt
Hom(M, N)
Hom(M, N) : ModRng, ModRng -> ModMatRng
AHom(M, N) : ModRng, ModRng -> ModMatRng
GHomOverCentralizingField(M, N) : ModGrp, ModGrp -> ModMatGrp
Example ModAlg_EndoRing (H86E11)
Example ModAlg_CreateHomGHom (H86E12)
Endo-- and Automorphisms
EndomorphismAlgebra(M) : ModRng -> AlgMat
CentreOfEndomorphismRing(M) : ModRng -> AlgMat
AutomorphismGroup(M) : ModRng -> GrpMat
IsIsomorphic(M, N) : ModRng, ModRng -> BoolElt, AlgMatElt
Example ModAlg_EndoRing (H86E13)
Modules over a General Algebra
Introduction
Construction of Algebra Modules
Module(A, m): Alg, Map[SetCart, ModRng] -> ModAlg
Example ModAlg_AlgModCreate (H86E14)
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 (H86E15)
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 (H86E16)
sub< M | S > : ModAlg, [ModAlgElt] -> ModAlg
quo< M | S > : ModAlg, [ModAlgElt] -> ModAlg
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