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INVARIANT THEORY

 
Acknowledgements
 
Introduction
 
Invariant Rings of Finite Groups
      Creation
      Access
 
Group Actions on Polynomials
 
Permutation Group Actions on Polynomials
 
Matrix Group Actions on Polynomials
 
Algebraic Group Actions on Polynomials
 
Verbosity
 
Construction of Invariants of Specified Degree
 
Construction of G-modules
 
Molien Series
 
Primary Invariants
 
Secondary Invariants
 
Fundamental Invariants
 
The Module of an Invariant Ring
 
The Algebra of an Invariant Ring and Algebraic Relations
 
Properties of Invariant Rings
 
Steenrod Operations
 
Minimalization and Homogeneous Module Testing
 
Attributes of Invariant Rings and Fields
 
Invariant Rings of Linear Algebraic Groups
      Creation
      Access
      Functions
 
Invariant Fields
      Creation
      Access
      Functions for Invariant Fields
 
Invariants of the Symmetric Group
 
Bibliography







DETAILS

 
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 (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)

 
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 (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 Wed Apr 24 15:09:57 EST 2013