Introduction

 
Reflexive Forms

      Quadratic Forms

 
Inner Products
      Example FldForms_generalform (H30E1)
      EnsureUpperTriangular(A) : AlgMatElt -> AlgMatElt
      DotProduct(u, v) : ModTupFldElt, ModTupFldElt -> FldElt
      DotProductMatrix(S) : SeqEnum[ModTupFldElt] -> AlgMatElt
      GramMatrix(V) : ModTupRng -> AlgMatElt
      InnerProductMatrix(V) : ModTupRng -> AlgMatElt
      Example FldForms_grammatrix (H30E2)
      Example FldForms_innerprod (H30E3)

      Orthogonality
            OrthogonalComplement(V, X : parameters) : ModTupFld, ModTupFld -> ModTupFld
            Radical(V : parameters) : ModTupFld -> ModTupFld
            IsNondegenerate(V) : ModTupFld -> BoolElt
            IsDegenerate(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
      IsTotallyIsotropic(V) : ModTupFld -> BoolElt
      IsTotallySingular(V) : ModTupFld -> BoolElt
      MaximalTotallyIsotropicSubspace(V) : ModTupFld -> ModTupFld
      MaximalTotallySingularSubspace(V) : ModTupFld -> ModTupFld
      WittIndex(V) : ModTupFld -> RngIntElt
      HyperbolicPair(V, u) : ModTupFld, ModTupFldElt -> ModTupFldElt
      Example FldForms_pseudoalt (H30E4)
      Example FldForms_quadsplit (H30E5)
      HyperbolicSplitting(V) : ModTupFld -> Tup
      Example FldForms_hypsplit (H30E6)
      Example FldForms_extradical (H30E7)
      SymplecticBasis(V,U,W) : ModTupFld, ModTupFld, ModTupFld -> [ModTupFldElt]
      WittDecomposition(V) : ModTupFld -> SeqEnum[ModTupFld]
      WittDecomposition(M, a) : AlgMatElt[FldFin], FldAut -> AlgMatElt[FldFin], AlgMatElt[FldFin]
      MetabolicSpace(V) : ModTupFld -> ModTupFld

 
The Standard Forms
      StandardAlternatingForm(n,R) : RngIntElt, Rng -> AlgMatElt
      Example FldForms_alternatingform (H30E8)
      StandardPseudoAlternatingForm(n,K) : RngIntElt, Fld -> AlgMatElt
      StandardHermitianForm(n,K) : RngIntElt, Fld -> AlgMatElt, Map
      StandardQuadraticForm(n, K : parameters) : RngIntElt, Fld -> AlgMatElt
      Example FldForms_minusform (H30E9)
      Example FldForms_revisedminus (H30E10)
      StandardSymmetricForm(n, K) : RngIntElt, Fld -> AlgMatElt

 
Constructing Polar Spaces
      PolarSpace(F, a) : AlgMatElt[FldFin], FldAut -> ModTupFld[FldFin]
      TrivialPolarSpace(F, n) : FldFin, RngIntElt -> ModTupFld[FldFin]
      IsPolarSpace(V) : ModTupFld -> BoolElt
      PolarSpaceType(V) : ModTupFld -> MonStgElt
      Example FldForms_polarspace (H30E11)

      Isotropic Subspaces
            FirstIsotropicSubspace(V, k) : ModTupFld[FldFin], RngIntElt -> ModTupFld[FldFin]
            NextIsotropicSubspace(V, k) : ModTupFld[FldFin], RngIntElt -> ModTupFld[FldFin]
            AllIsotropicSubspaces(V, k) : ModTupFld[FldFin], RngIntElt -> [ ModTupFld[FldFin] ]
            NumberOfIsotropicSubspaces(V, k) : ModTupFld[FldFin], RngIntElt -> RngIntElt

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

      Unitary Spaces
            UnitarySpace(J, sigma) : AlgMatElt, Map -> ModTupFld
            IsUnitarySpace(W) : ModTupFld -> BoolElt, RngIntElt
            Example FldForms_unitaryform (H30E12)
            ConjugateTranspose(M, sigma) : Mtrx, Map -> Mtrx

      Quadratic Spaces
            QuadraticSpace(Q) : AlgMatElt -> ModTupRng
            QuadraticSpace(f) : RngMPolElt -> ModTupRng
            SymmetricToQuadraticForm(J) : AlgMatElt -> AlgMatElt
            QuadraticFormMatrix(V) : ModTupRng -> ModAlgElt
            QuadraticNorm(v) : ModTupFldElt -> FldElt
            QuadraticFormPolynomial(V) : ModTupRng -> RngPolElt
            QuadraticFormPolynomial(Q) : AlgMatElt -> RngPolElt
            Example FldForms_polyquad (H30E13)
            OrthogonalSum(V, W) : ModTupFld, ModTupFld -> ModTupFld, Map, Map
            OrthogonalTensorProduct(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
            SpinorNorm(Q, g) : AlgMatElt[Fld], AlgMatElt[Fld] -> FldElt
            Example FldForms_spinor-general (H30E14)
            CharacterQQModSquares(d, r) : RngIntElt, FldRatElt -> RngIntElt
            SpinorNormRho(d, g, Q) : RngIntElt, GrpMatElt, AlgMatElt -> 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 (H30E15)

 
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 (H30E16)
            Example FldForms_transform (H30E17)
            Example FldForms_transformalt (H30E18)
            CommonComplement(V, U, W) : ModTupFld, ModTupFld, ModTupFld -> ModTupFld
            ExtendIsometry(V, U, f) : ModTupFld, ModTupFld, Map -> Map
            IsometryGroup(V) : ModTupFld -> GrpMat
            Example FldForms_isometrygroup (H30E19)
            Example FldForms_conjisom (H30E20)

      Similarities
            IsSimilarity(U, V, f) : ModTupFld, ModTupFld, Map -> BoolElt, FldElt
            IsSimilarity(f) : Map -> BoolElt, FldElt
            IsSimilarity(V, g) : ModTupFld, Mtrx -> BoolElt, FldElt
            IsSimilar(V, W) : ModTupFld, ModTupFld -> BoolElt, Map
            Example FldForms_simherm (H30E21)
            SimilarityGroup(V) : ModTupFld -> GrpMat

      Gram-Schmidt Normalisation
            GramSchmidtPair(J) : AlgMatElt -> AlgMatElt, AlgMatElt
            Example FldForms_hermitiangs (H30E22)
            Example FldForms_skewgs (H30E23)

 
Classical Groups
      Example FldForms_fixaltform (H30E24)
      Example FldForms_fixhermform (H30E25)

 
Lie Algebras and Bilinear Forms
      DerivationAlgebra(J) : AlgMatElt -> AlgLie
      Example FldForms_lieC3 (H30E26)
      Example FldForms_lieG2 (H30E27)
      HeisenbergAlgebra(J) : AlgMatElt -> AlgLie
      Example FldForms_heisenberg (H30E28)

 
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
      GeneralisedWallForm(V, f) : ModTupFld, Mtrx -> ModTupFld, Map

 
Invariant Forms
      InvariantBilinearForms(G) : GrpMat -> SeqEnum[AlgMatElt], SeqEnum[AlgMatElt]
      Example FldForms_reducible (H30E29)
      Example FldForms_nonabs (H30E30)
      InvariantQuadraticForms(G) : GrpMat -> SeqEnum[AlgMatElt]
      Example FldForms_invquadform (H30E31)
      SemilinearDual(M, mu) : ModGrp,Map -> ModGrp
      InvariantSesquilinearForms(G) : GrpMat -> SeqEnum[AlgMatElt]
      Example FldForms_sesquiforms (H30E32)
      Example FldForms_hermandalt (H30E33)
      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 (H30E34)

 
Polar Spaces More Generally

      Creation of PolarSpaces
            AmbientPolarSpace(J) : AlgMatElt -> SpcPlr
            PolarSpace(L) : LatNF -> SpcPlr
            ChangeRing(V, R) : SpcPlr, Rng -> SpcPlr

      Properties of Polar Spaces
            BaseField(V) : SpcPlr -> Fld
            VectorSpace(V) : SpcPlr -> ModTupFld
            Dimension(V) : SpcPlr -> RngIntElt
            InnerForm(V) : SpcPlr -> AlgMatElt
            Involution(V) : SpcPlr -> FldAut
            SpaceType(V) : SpcPlr -> MonStgElt
            Diagonal(V) : SpcPlr -> AlgMatElt

      Predicates on Polar Spaces
            IsDefinite(V) : SpcPlr -> BoolElt

 
Bibliography

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