Invariant Theory

• Introduction
• Invariant Rings of Finite Groups
  • Creation
    • InvariantRing(G): GrpMat → RngInvar
    • InvariantRing(G, K): GrpPerm, Fld → RngInvar
  • Access
    • Group(R): RngInvar → Grp
    • CoefficientRing(R): RngInvar → Grp
    • CoefficientField(R): RngInvar → Rng
    • 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: Group Actions
• 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(G, d): GrpMat, RngIntElt → [ RngMPolElt ]
  • InvariantsOfDegree(G, K, d): GrpPerm, Fld, RngIntElt → [ RngMPolElt ]
  • InvariantsOfDegree(G, P, d): GrpMat, RngMPol, RngIntElt → [ RngMPolElt ]
  • InvariantsOfDegree(G, P, d): GrpPerm, RngMPol, RngIntElt → [ RngMPolElt ]
  • InvariantsOfDegree(R, d, k): RngInvar, RngIntElt, RngIntElt → [ RngMPolElt ]
  • InvariantsOfDegree(G, d, k): GrpMat, RngIntElt, RngIntElt → [ RngMPolElt ]
  • InvariantsOfDegree(G, K, d, k): GrpPerm, Fld, RngIntElt, RngIntElt → [ RngMPolElt ]
  • InvariantsOfDegree(G, P, d, k): GrpPerm, RngMPol, RngIntElt, RngIntElt → [ RngMPolElt ]
  • Example: Invariants Of Degree
  • SetAllInvariantsOfDegree(R, d, Q): RngInvar, RngIntElt, [ RngMPolElt ]
  • Example: Invariants Of Degree
• 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: G Module
• Molien Series
  • MolienSeries(G): GrpMat → FldFunUElt
  • MolienSeries(G): GrpPerm → FldFunUElt
  • MolienSeriesApproximation(G, n): GrpPerm, RngIntElt → RngSerLaurElt
  • Example: Molien Series
• Primary Invariants
  • PrimaryInvariants(R): RngInvar → [ RngMPolElt ]
  • Example: Adem Milgram
• Secondary Invariants
  • SecondaryInvariants(R): RngInvar → [ RngMPolElt ]
  • SecondaryInvariants(R, H): RngInvar, Grp → [ RngMPolElt ]
  • IrreducibleSecondaryInvariants(R): RngInvar → [ RngMPolElt ]
  • Example: Secondary Invariants
• Fundamental Invariants
  • FundamentalInvariants(R): RngInvar → [ RngMPolElt ]
  • Example: Fundamental Invariants
  • Example: Transitive Groups Degree7
  • Example: S5Degree10
• The Module of an Invariant Ring
  • Module(R): RngInvar → ModMPol, Map
  • Example: Module
• 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: Relations
• 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: Depth
• Steenrod Operations
  • SteenrodOperation(f, i): RngMPolElt, RngIntElt → RngMPolElt
  • Example: Steenrod Operation
• Minimalization and Homogeneous Module Testing
  • MinimalAlgebraGenerators(L): [ RngMPol ] → [ RngMPol ]
  • MinimalAlgebraGenerators(L): { RngMPol } → [ RngMPol ]
  • HomogeneousModuleTest(P, S, F): [ RngMPol ], [ RngMPol ], RngMPol → BoolElt, [ RngMPol ]
  • HomogeneousModuleTest(P, S, L): [ RngMPol ], [ RngMPol ], [ RngMPol ] → [ Bool Elt ], [ [ RngMPol ] ]
  • Example: Minimal Algebra Generators
  • Example: Homogeneous Module Test2
• Attributes of Invariant Rings and Fields
  • R`PrimaryInvariants
  • R`SecondaryInvariants
  • R`HilbertSeries
  • Example: Attributes
• Invariant Rings of Linear Algebraic Groups
  • Creation
    • InvariantRing(I, A): RngMPol, Mtrx → RngInvar
    • BinaryForms(N, p): [RngIntElt], RngIntElt → RngMPol, [[RngMPolElt]], RngMPol
    • 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: SL2 Invar
    • Example: SL2 Tensor
    • Example: Alg Group1
    • Example: Alg Group2
• Invariant Fields
  • Creation
    • InvariantField(G, K): GrpPerm, Fld → FldInvar
    • InvariantField(G): GrpMat → RngInvar
    • InvariantField(I, A): RngMPol, Mtrx → 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: Invar Field1
    • Example: Invar Field2
• Invariants of the Symmetric Group
  • ElementarySymmetricPolynomial(P, k): RngMPol, RngIntElt → RngMPolElt
  • IsSymmetric(f): RngMPolElt → BoolElt, RngMPolElt
  • IsSymmetric(f, S): RngMPolElt, RngMPol → BoolElt, RngMPolElt
  • Example: Is Symmetric