Introduction

 
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

 
Group Actions on Polynomials

 
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 (H120E1)

 
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 (H120E2)
      SetAllInvariantsOfDegree(R, d, Q) : RngInvar, RngIntElt, [ RngMPolElt ] ->
      Example RngInvar_InvariantsOfDegree (H120E3)

 
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 (H120E4)

 
Molien Series
      MolienSeries(G) : GrpMat -> FldFunUElt
      MolienSeriesApproximation(G, n) : GrpPerm, RngIntElt -> RngSerLaurElt
      Example RngInvar_MolienSeries (H120E5)

 
Primary Invariants
      PrimaryInvariants(R) : RngInvar -> [ RngMPolElt ]
      Example RngInvar_AdemMilgram (H120E6)

 
Secondary Invariants
      SecondaryInvariants(R) : RngInvar -> [ RngMPolElt ]
      SecondaryInvariants(R, H) : RngInvar, Grp -> [ RngMPolElt ]
      IrreducibleSecondaryInvariants(R) : RngInvar -> [ RngMPolElt ]
      Example RngInvar_SecondaryInvariants (H120E7)

 
Fundamental Invariants
      FundamentalInvariants(R) : RngInvar -> [ RngMPolElt ]
      Example RngInvar_FundamentalInvariants (H120E8)
      Example RngInvar_TransitiveGroupsDegree7 (H120E9)
      Example RngInvar_S5Degree10 (H120E10)

 
The Module of an Invariant Ring
      Module(R) : RngInvar -> ModMPol, Map
      Example RngInvar_Module (H120E11)

 
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 (H120E12)

 
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 (H120E13)

 
Steenrod Operations
      SteenrodOperation(f, i) : RngMPolElt, RngIntElt -> RngMPolElt
      Example RngInvar_SteenrodOperation (H120E14)

 
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 (H120E15)
      Example RngInvar_HomogeneousModuleTest2 (H120E16)

 
Attributes of Invariant Rings and Fields
      R`PrimaryInvariants
      R`SecondaryInvariants
      R`HilbertSeries
      Example RngInvar_Attributes (H120E17)

 
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 (H120E18)
            Example RngInvar_SL2-tensor (H120E19)
            Example RngInvar_AlgGroup1 (H120E20)
            Example RngInvar_AlgGroup2 (H120E21)

 
Invariant Fields

      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 (H120E22)
            Example RngInvar_InvarField2 (H120E23)

 
Invariants of the Symmetric Group
      ElementarySymmetricPolynomial(P, k) : RngMPol, RngIntElt -> RngMPolElt
      IsSymmetric(f) : RngMPolElt -> BoolElt, RngMPolElt
      Example RngInvar_IsSymmetric (H120E24)

 
Bibliography

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