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POLAR SPACES

 
Acknowledgements
 
Introduction
 
Reflexive Forms
      Quadratic Forms
 
Inner Products
      Orthogonality
 
Isotropic and Singular Vectors and Subspaces
 
The Standard Forms
 
Constructing Polar Spaces
      Symplectic Spaces
      Unitary Spaces
      Quadratic Spaces
 
Isometries and Similarities
      Isometries
      Similarities
 
Wall Forms
 
Invariant Forms
      Semi-invariant Forms
 
Bibliography







DETAILS

 
Introduction

 
Reflexive Forms

      Quadratic Forms

 
Inner Products
      Example FldForms_generalform (H29E1)
      DotProduct(u, v) : ModTupFldElt, ModTupFldElt -> FldElt
      DotProductMatrix(W) : SeqEnum[ModTupFldElt] -> AlgMatElt
      GramMatrix(V) : ModTupRng -> AlgMatElt
      InnerProductMatrix(V) : ModTupRng -> AlgMatElt
      Example FldForms_grammatrix (H29E2)
      Example FldForms_innerprod (H29E3)

      Orthogonality
            OrthogonalComplement(V, X : parameters) : ModTupFld, ModTupFld -> ModTupFld
            Radical(V : parameters) : ModTupFld -> ModTupFld
            IsNondegenerate(V) : ModTupFld -> BoolElt
            SingularRadical(V) : ModTupFld -> ModTupFld
            IsNonsingular(V) : ModTupFld -> BoolElt

 
Isotropic and Singular Vectors and Subspaces
      HasIsotropicVector(V) : ModTupFld -> BoolElt, ModTupFldElt
      HasSingularVector(V) : ModTupFld -> BoolElt, ModTupFldElt
      HyperbolicPair(V, u) : ModTupFld, ModTupFldElt -> ModTupFldElt
      Example FldForms_pseudoalt (H29E4)
      HyperbolicSplitting(V) : ModTupFld -> SeqEnum, SeqEnum
      Example FldForms_hypsplit (H29E5)
      Example FldForms_extradical (H29E6)
      IsTotallyIsotropic(V) : ModTupFld) -> BoolElt
      IsTotallySingular(V) : ModTupFld) -> BoolElt
      WittDecomposition(V) : ModTupFld -> SeqEnum[ModTupFld]
      WittIndex(V) : ModTupFld -> RngIntElt
      MaximalTotallyIsotropicSubspace(V) : ModTupFld -> ModTupFld
      MaximalTotallySingularSubspace(V) : ModTupFld -> ModTupFld

 
The Standard Forms
      StandardAlternatingForm(n,R) : RngIntElt, Rng -> AlgMatElt
      Example FldForms_alternatingform (H29E7)
      StandardPseudoAlternatingForm(n,K) : RngIntElt, Fld -> AlgMatElt
      StandardHermitianForm(n,K) : RngIntElt, Fld -> AlgMatElt, Map
      StandardQuadraticForm(n, K : parameters) : RngIntElt, Fld -> AlgMatElt
      Example FldForms_minusform (H29E8)
      Example FldForms_revisedminus (H29E9)
      StandardSymmetricForm(n, K) : RngIntElt, Fld -> AlgMatElt

 
Constructing Polar Spaces
      IsPolarSpace(V) : ModTupFld -> BoolElt
      PolarSpaceType(V) : ModTupFld -> MonStgElt
      Example FldForms_polarspace (H29E10)

      Symplectic Spaces
            SymplecticSpace(J) : AlgMatElt -> ModTupRng
            IsSymplecticSpace(W) : ModTupFld -> BoolElt
            IsPseudoSymplecticSpace(W) : ModTupFld -> BoolElt

      Unitary Spaces
            UnitarySpace(J, sigma) : AlgMatElt, Map -> ModTupFld
            IsUnitarySpace(W) : ModTupFld -> BoolElt
            Example FldForms_unitaryform (H29E11)
            ConjugateTranspose(M, sigma) : Mtrx, Map -> Mtrx

      Quadratic Spaces
            QuadraticSpace(Q) : AlgMatElt -> ModTupFld
            QuadraticSpace(f) : RngMPolElt -> ModTupFld
            SymmetricToQuadraticForm(J) : AlgMatElt -> AlgMatElt
            QuadraticFormMatrix(V) : ModTupRng -> ModAlgElt
            QuadraticNorm(v) : ModTupFldElt -> FldElt
            QuadraticFormPolynomial(V) : ModTupRng -> RngPolElt
            Example FldForms_polyquad (H29E12)
            OrthogonalSum(V, W) : ModTupFld, ModTupFld) -> ModTupFld
            TotallySingularComplement(V, U, W) : ModTupFld, ModTupFld, ModTupFld) -> ModTupFld
            Discriminant(V) : ModTupFld -> RngIntElt
            ArfInvariant(V) : ModTupFld -> RngIntElt
            DicksonInvariant(V, f) : ModTupFld, Mtrx -> RngIntElt
            SpinorNorm(V, f) : ModTupFld, Mtrx -> RngIntElt
            HyperbolicBasis(U, B, W) : ModTupFld, SeqEnum, ModTupFld -> SeqEnum
            OrthogonalReflection(a) : ModTupFldElt -> AlgMatElt
            RootSequence(V, f) : ModTupFld, Mtrx -> SeqEnum
            ReflectionFactors(V, f) : ModTupFld, Mtrx) -> SeqEnum
            SiegelTransformation(u, v) : ModTupFldElt, ModTupFldElt -> AlgMatElt
            Example FldForms_siegel (H29E13)

 
Isometries and Similarities

      Isometries
            IsIsometry(U, V, f) : ModTupFld, ModTupFld, Map -> BoolElt
            IsIsometry(f) : Map -> BoolElt
            IsIsometry(V, g) : ModTupFld, Mtrx -> BoolElt
            IsIsometric(V, W) : ModTupFld, ModTupFld -> BoolElt, Map
            Example FldForms_isometric (H29E14)
            Example FldForms_transform (H29E15)
            Example FldForms_transformalt (H29E16)
            CommonComplement(V, U, W) : ModTupFld, ModTupFld, ModTupFld) -> ModTupFld
            ExtendIsometry(V, U, f) : ModTupFld, ModTupFld, Map -> Map
            IsometryGroup(V) : ModTupFld) -> GrpMat
            Example FldForms_isometrygroup (H29E17)
            Example FldForms_conjisom (H29E18)

      Similarities
            IsSimilarity(U, V, f) : ModTupFld, ModTupFld, Map -> BoolElt
            IsSimilarity(f) : Map -> BoolElt
            IsSimilarity(V, g) : ModTupFld, Mtrx -> BoolElt
            SimilarityGroup(V) : ModTupFld) -> GrpMat

 
Wall Forms
      WallForm(V, f) : ModTupFld, Mtrx -> ModTupFld, Map
      WallIsometry(V, I, mu) : ModTupFld, ModTupFld, Map -> Mtrx
      WallDecomposition(V, f) : ModTupFld, Mtrx -> Mtrx, Mtrx
      SemiOrthogonalBasis(V) : ModTupFld) -> SeqEnum

 
Invariant Forms
      InvariantBilinearForms(G) : GrpMat -> SeqEnum[AlgMatElt], SeqEnum[AlgMatElt]
      Example FldForms_reducible (H29E19)
      Example FldForms_nonabs (H29E20)
      InvariantQuadraticForms(G) : GrpMat -> SeqEnum[AlgMatElt]
      Example FldForms_invquadform (H29E21)
      SemilinearDual(M, mu) : ModGrp,Map -> ModGrp
      InvariantSesquilinearForms(G) : GrpMat -> SeqEnum[AlgMatElt]
      Example FldForms_sesquiforms (H29E22)
      Example FldForms_hermandalt (H29E23)
      InvariantFormBases(G) : GrpMat -> SeqEnum[AlgMatElt], SeqEnum[AlgMatElt], SeqEnum[AlgMatElt], SeqEnum[AlgMatElt]

      Semi-invariant Forms
            TwistedDual(M, lambda) : ModGrp, Map -> ModGrp
            SemiInvariantBilinearForms(G) : GrpMat -> SeqEnum
            SemiInvariantQuadraticForms(G) : GrpMat -> SeqEnum
            TwistedSemilinearDual(M, lambda, mu) : ModGrp, Map, Map -> ModGrp
            SemiInvariantSesquilinearForms(G) : GrpMat -> SeqEnum
            Example FldForms_semiinv (H29E24)

 
Bibliography

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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013