General Constructions
RModule(A) : AlgMat -> ModRng
RModule(Q) : [ MtrxS ] -> ModTupRng
Example ModAlg_CreateK6 (H89E1)
Constructions for K[G]-Modules
GModule(G, Q) : Grp, [ MtrxS ] -> ModGrp
PermutationModule(G, K) : GrpPerm, Fld -> ModGrp
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 (H89E2)
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 }
Construction
sub<M | L> : ModRng, List -> ModRng
ImageWithBasis(X, M) : ModMatRngElt, ModRng -> ModRng
Morphism(M, N) : ModRng, ModRng -> ModMatRngElt
Example ModAlg_Submodule (H89E3)
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 (H89E4)
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 (H89E5)
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 (H89E6)
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 (H89E7)
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 (H89E8)
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 (H89E9)
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 (H89E10)
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 (H89E11)
Example ModAlg_CreateHomGHom (H89E12)
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 (H89E13)
Modules over a General Algebra
Construction of Algebra Modules
Module(A, m): Alg, Map[SetCart, ModRng] -> ModAlg
Example ModAlg_AlgModCreate (H89E14)
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 (H89E15)
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 (H89E16)
sub< M | S > : ModAlg, [ModAlgElt] -> ModAlg
quo< M | S > : ModAlg, [ModAlgElt] -> ModAlg
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Version: V2.19 of
Mon Dec 17 14:40:36 EST 2012