Invariant Rings of Finite Groups
Creation
InvariantRing(G) : GrpMat -> RngInvar
Access
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 (H110E1)
Algebraic Group Actions on Polynomials
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 (H110E2)
SetAllInvariantsOfDegree(R, d, Q) : RngInvar, RngIntElt, [ RngMPolElt ] ->
Example RngInvar_InvariantsOfDegree (H110E3)
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 (H110E4)
Molien Series
MolienSeries(G) : GrpMat -> FldFunUElt
MolienSeriesApproximation(G, n) : GrpPerm, RngIntElt -> RngSerLaurElt
Example RngInvar_MolienSeries (H110E5)
Primary Invariants
PrimaryInvariants(R) : RngInvar -> [ RngMPolElt ]
Example RngInvar_AdemMilgram (H110E6)
Secondary Invariants
SecondaryInvariants(R) : RngInvar -> [ RngMPolElt ]
SecondaryInvariants(R, H) : RngInvar, Grp -> [ RngMPolElt ]
IrreducibleSecondaryInvariants(R) : RngInvar -> [ RngMPolElt ]
Example RngInvar_SecondaryInvariants (H110E7)
Fundamental Invariants
FundamentalInvariants(R) : RngInvar -> [ RngMPolElt ]
Example RngInvar_FundamentalInvariants (H110E8)
Example RngInvar_TransitiveGroupsDegree7 (H110E9)
Example RngInvar_S5Degree10 (H110E10)
The Module of an Invariant Ring
Module(R) : RngInvar -> ModMPol, Map
Example RngInvar_Module (H110E11)
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 (H110E12)
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 (H110E13)
Steenrod Operations
SteenrodOperation(f, i) : RngMPolElt, RngIntElt -> RngMPolElt
Example RngInvar_SteenrodOperation (H110E14)
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 (H110E15)
Example RngInvar_HomogeneousModuleTest2 (H110E16)
Attributes of Invariant Rings and Fields
R`PrimaryInvariants
R`SecondaryInvariants
R`HilbertSeries
Example RngInvar_Attributes (H110E17)
Invariant Rings of Linear Algebraic Groups
Creation
InvariantRing(I, A) : RngMPol, Mtrx -> RngInvar
BinaryForms(N, p) : [RngIntElt], RngIntElt -> RngMPol, [[RngMPolElt]], RngMPol
Access
GroupIdeal(R) : RngInvar -> RngMPol
Representation(R) : RngInvar -> Mtrx
Functions
InvariantsOfDegree(R, d) : RngInvar, RngIntElt -> [ RngMPolElt ]
FundamentalInvariants(R) : RngInvar -> RngMPol
DerksenIdeal(R) : RngInvar -> [RngMPolElt]
HilbertIdeal(R) : RngInvar -> RngMPol
Example RngInvar_SL2-invar (H110E18)
Example RngInvar_SL2-tensor (H110E19)
Example RngInvar_AlgGroup1 (H110E20)
Example RngInvar_AlgGroup2 (H110E21)
Creation
InvariantField(G, K) : GrpPerm, Fld -> FldInvar
Access
FunctionField(F) : FldInvar -> FldFunRat
Group(F) : FldInvar -> Grp
GroupIdeal(F) : FldInvar -> RngMPol
Representation(F) : FldInvar -> Mtrx
Functions for Invariant Fields
FundamentalInvariants(F) : FldInvar -> RngMPol
DerksenIdeal(F) : FldInvar -> RngMPol
MinimizeGenerators(L) : [FldFunRatElt] -> [FldFunRatElt]
QuadeIdeal(L) : [FldFunRatElt] -> RngMPol
Example RngInvar_InvarField1 (H110E22)
Example RngInvar_InvarField2 (H110E23)
Invariants of the Symmetric Group
ElementarySymmetricPolynomial(P, k) : RngMPol, RngIntElt -> RngMPolElt
IsSymmetric(f) : RngMPolElt -> BoolElt, RngMPolElt
Example RngInvar_IsSymmetric (H110E24)
Bibliography
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Version: V2.19 of
Mon Dec 17 14:40:36 EST 2012