Creation of Invariant Rings
InvariantRing(G) : GrpMat -> RngInvar
Accessing Invariant Rings
Group(R) : RngInvar -> Grp
CoefficientRing(R) : RngInvar -> Grp
PolynomialRing(R) : RngInvar -> RngMPol
f in R : RngMPol, RngInvar -> FldFunUElt, ModMPolElt
Permutation Group Actions on Polynomials
f ^ g : RngMPolElt, GrpPermElt -> RngMPolElt
f ^ G : RngMPolElt, GrpPerm -> { RngMPolElt }
IsInvariant(f, g) : RngMPolElt, GrpElt -> BoolElt
IsInvariant(f, G) : RngMPolElt, Grp -> BoolElt
Matrix Group Actions on Polynomials
f ^ a : RngMPolElt, GrpMatElt -> RngMPolElt
f ^ G : RngMPolElt, GrpMat -> { RngMPolElt }
Example RngInvar_GroupActions (H81E1)
Construction of G-modules
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 RngInvar_GModule (H81E2)
Verbosity
SetVerbose("Invariants", v) : MonStgElt, RngIntElt ->
Construction of Invariants of Specified Degree
ReynoldsOperator(f, G) : RngMPolElt, GrpMat -> RngMPolElt
InvariantsOfDegree(R, d) : RngInvar, RngIntElt -> [ RngMPolElt ]
InvariantsOfDegree(R, d, k) : RngInvar, RngIntElt, RngIntElt -> [ RngMPolElt ]
Example RngInvar_InvariantsOfDegree (H81E3)
SetAllInvariantsOfDegree(R, d, Q) : RngInvar, RngIntElt, [ RngMPolElt ] ->
Example RngInvar_InvariantsOfDegree (H81E4)
Molien Series
MolienSeries(G) : GrpMat -> FldFunUElt
MolienSeriesApproximation(G, n) : GrpPerm, RngIntElt -> RngSerLaurElt
Example RngInvar_MolienSeries (H81E5)
Primary Invariants
PrimaryInvariants(R) : RngInvar -> [ RngMPolElt ]
Example RngInvar_AdemMilgram (H81E6)
Secondary Invariants
SecondaryInvariants(R) : RngInvar -> [ RngMPolElt ]
SecondaryInvariants(R, H) : RngInvar, Grp -> [ RngMPolElt ]
IrreducibleSecondaryInvariants(R) : RngInvar -> [ RngMPolElt ]
Example RngInvar_SecondaryInvariants (H81E7)
Fundamental Invariants
FundamentalInvariants(R) : RngInvar -> [ RngMPolElt ]
Example RngInvar_FundamentalInvariants (H81E8)
The Module of an Invariant Ring
Module(R) : RngInvar -> ModMPol, Map
Example RngInvar_Module (H81E9)
The Algebra of an Invariant Ring and Algebraic Relations
Algebra(R) : RngInvar -> RngMPol, [ RngMPolElt ]
Relations(R) : RngInvar -> [ RngMPolElt ]
RelationIdeal(R) : RngInvar -> RngMPol
PrimaryAlgebra(R) : RngInvar -> RngMPol
PrimaryIdeal(R) : RngInvar -> RngMPol
Example RngInvar_Relations (H81E10)
Properties of Invariant Rings
HilbertSeries(R) : RngInvar -> FldFunUElt
HilbertSeriesApproximation(R, n) : RngInvar, RngIntElt -> RngSerLaurElt
IsCohenMacaulay(R) : RngInvar -> BoolElt
FreeResolution(R) : RngInvar -> [ ModMPol ]
MinimalFreeResolution(R) : RngInvar -> [ ModMPol ]
HomologicalDimension(R) : RngInvar -> RngInt
Depth(R) : RngInvar -> RngIntElt
Example RngInvar_Depth (H81E11)
Steenrod Operations
SteenrodOperation(f, i) : RngMPolElt, RngIntElt -> RngMPolElt
Example RngInvar_SteenrodOperation (H81E12)
Minimalization and Homogeneous Module Testing
MinimalAlgebraGenerators(L) : [ RngMPol ] -> [ RngMPol ]
HomogeneousModuleTest(P, S, F) : [ RngMPol ], [ RngMPol ], RngMPol -> BoolElt, [ RngMPol ]
HomogeneousModuleTest(P, S, L) : [ RngMPol ], [ RngMPol ], [ RngMPol ] -> [ BoolElt ], [ [ RngMPol ] ]
Example RngInvar_MinimalAlgebraGenerators (H81E13)
Example RngInvar_HomogeneousModuleTest2 (H81E14)
Attributes of Invariant Rings
R`PrimaryInvariants
R`SecondaryInvariants
R`HilbertSeries