General Constructions
RModule(A) : AlgMat -> ModTupRng
RModule(Q) : [ AlgMatElt ] -> ModTupRng
Example ModAlg_CreateK6 (H71E1)
Constructions for K[G]-Modules
GModule(G, Q) : Grp, [ GrpMatElt ] -> ModGrp
PermutationModule(G, K) : Grp, Fld -> ModGrp
The Underlying Vector Space
VectorSpace(M) : ModTupRng -> ModTupRng
M . i : ModTupRng, RngIntElt -> ModElt
CoefficientRing(M) : ModTupRng -> Rng
Generators(M) : ModTupRng -> { ModTupElt }
Parent(u) : ModTupElt -> ModRng
The Algebra
Algebra(M) : ModTupRng -> Rng
Action(M) : ModTupRng -> AlgMat
MatrixGroup(M) : ModGrp -> GrpMat
ActionGenerator(M, i) : ModTupRng, RngIntElt -> AlgMatElt
NumberOfActionGenerators(M) : ModTupRng -> RngIntElt
Group(M) : ModGrp -> Grp
Example ModAlg_Access (H71E2)
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 : ModGrp, AlgMatElt -> ModGrp
Element Construction and Operations
Construction of Module Elements
elt< M | a_1, ..., a_n > : ModTupRng, List -> ModTupRngElt
M ! Q : ModTupRng, [RngElt] -> ModTupRngElt
Zero(M) : ModRng, RngIntElt -> ModRngElt
Random(M) : ModRng -> ModRngElt
Deconstruction of Module Elements
ElementToSequence(u) : ModTupRngElt -> [RngElt]
Action of the Algebra on the Module
u * a : ModTupElt, AlgElt -> ModTupElt
u * g : ModGrpElt, GrpElt -> ModGrpElt
Arithmetic with Module Elements
u + v : ModTupElt, ModTupElt -> ModTupElt
- u : ModTupElt -> ModTupElt
u - v : ModTupElt, ModTupElt -> ModTupElt
k * u : RngElt, ModTupElt -> ModTupElt
u * k : ModTupElt, RngElt -> ModTupElt
u / k : ModTupElt, RngElt -> ModTupElt
Indexing
u[i] : ModTupRngElt, RngIntElt -> RngElt
u[i] := x : ModTupRngElt, RngIntElt, RngElt -> ModTupRngElt
Properties of Module Elements
IsZero(u) : ModTupElt -> BoolElt
Support(u) : ModTupRngElt -> { RngElt }
Construction
sub<M | L> : ModTupRng, List -> ModTupRng
ImageWithBasis(X, M) : ModMatRngElt, ModRng -> ModRng
Morphism(M, N) : ModRng, ModRng -> ModMatRngElt
Example ModAlg_Submodule (H71E3)
Membership and Equality
u in M : ModTupRngElt, ModTupRng -> BoolElt
N subset M : ModTupRng, ModTupRng -> BoolElt
N eq M : ModTupRng, ModTupRng -> BoolElt
Operations on Submodules
M + N : ModTupRng, ModTupRng -> ModTupRng
M meet N : ModTupRng, ModTupRng -> ModTupRng
Quotient Modules
quo<M | L> : ModTupRng, List -> ModTupRng
Morphism(M, N) : ModRng, ModRng -> ModMatRngElt
Example ModAlg_QuotientModule (H71E4)
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 (H71E5)
MinimalField(M) : ModRng -> FldFin
Composition Series
CompositionSeries(M) : ModRng, ModRng -> [ ModRng ], [ ModRng ], AlgMatElt
CompositionFactors(M) : ModRng -> [ ModRng ]
Constituents(M) : ModRng -> [ ModRng ]
ConstituentsWithMultiplicities(M) : ModRng -> [ <ModRng, RngIntElt> ]
Example ModAlg_CompSeries (H71E6)
Socle Series
IsSemisimple(M) : ModGrp -> BoolElt
MaximalSubmodules(M) : ModRng -> [ ModRng ], BoolElt
JacobsonRadical(M) : ModRng -> ModRng
MinimalSubmodules(M) : ModRng -> [ ModRng ], BoolElt
MinimalSubmodules(M, F) : ModRng, ModRng -> [ ModRng ], BoolElt
MinimalSubmodule(M) : ModRng -> ModRng
Socle(M) : ModRng -> ModRng
SocleSeries(M) : ModRng -> [ ModRng ], [ ModRng ], AlgMatElt
SocleFactors(M) : ModRng -> [ ModRng ]
Example ModAlg_Minimals (H71E7)
Decomposabilty and Complements
IsDecomposable(M) : ModRng -> BoolElt, ModRng, ModRng
IndecomposableSummands(M) : ModGrp -> [ ModGrp ]
HasComplement(M, S) : ModGrp, ModGrp -> BoolElt, ModGrp
Complements(M, S) : ModGrp, ModGrp -> [ ModGrp ]
Example ModAlg_Decomposable (H71E8)
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 (H71E9)
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 : 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 (H71E10)
Creating Homomorphisms
hom< M -> N | X > : ModRng, ModRng, ModMatElt -> ModMatRng
H ! f : ModMatRng, Map -> ModMatRngElt
IsModuleHomomorphism(X) : ModMatElt -> BoolElt
Hom(M, N)
Hom(M, N) : ModRng, ModRng -> ModMatRng
AHom(M, N) : ModGrp, ModGrp -> ModMatGrp
GHomOverCentralizingField(M, N) : ModGrp, ModGrp -> ModMatGrp
Example ModAlg_EndoRing (H71E11)
Example ModAlg_CreateHomGHom (H71E12)
Endo-- and Automorphisms
EndomorphismAlgebra(M) : ModRng -> AlgMat
AutomorphismGroup(M) : ModRng -> AlgMat
IsIsomorphic(M, N) : ModRng, ModRng -> BoolElt, AlgMatElt
Example ModAlg_EndoRing (H71E13)