Polar Spaces

• Introduction
• Reflexive Forms
  • Quadratic Forms
• Inner Products
  • Example: generalform
  • EnsureUpperTriangular(A): AlgMatElt → AlgMatElt
  • DotProduct(u, v): ModTupFldElt, ModTupFldElt → FldElt
  • DotProductMatrix(S): SeqEnum[ModTupFldElt] → AlgMatElt
  • GramMatrix(V): ModTupRng → AlgMatElt
  • InnerProductMatrix(V): ModTupRng → AlgMatElt
  • Example: grammatrix
  • Example: innerprod
  • 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: pseudoalt
  • Example: quadsplit
  • HyperbolicSplitting(V): ModTupFld → Tup
  • Example: hypsplit
  • Example: extradical
  • 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
  • StandardAlternatingForm(n,q): RngIntElt, RngIntElt → AlgMatElt
  • Example: alternatingform
  • StandardPseudoAlternatingForm(n,K): RngIntElt, Fld → AlgMatElt
  • StandardPseudoAlternatingForm(n,q): RngIntElt, RngIntElt → AlgMatElt
  • StandardHermitianForm(n,K): RngIntElt, Fld → AlgMatElt, Map
  • StandardHermitianForm(n,q): RngIntElt, RngIntElt → AlgMatElt, Map
  • StandardQuadraticForm(n, K : parameters): RngIntElt, Fld → AlgMatElt
  • StandardQuadraticForm(n, q : parameters): RngIntElt, RngIntElt → AlgMatElt
  • Example: minusform
  • Example: revisedminus
  • StandardSymmetricForm(n, K): RngIntElt, Fld → AlgMatElt
  • StandardSymmetricForm(n, q : parameters): RngIntElt, RngIntElt → 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: polarspace
  • 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: unitaryform
    • 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: polyquad
    • 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: Spinor General
    • 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: siegel
• 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: isometric
    • Example: transform
    • Example: transformalt
    • CommonComplement(V, U, W): ModTupFld, ModTupFld, ModTupFld → ModTupFld
    • ExtendIsometry(V, U, f): ModTupFld, ModTupFld, Map → Map
    • IsometryGroup(V): ModTupFld → GrpMat
    • Example: isometrygroup
    • Example: conjisom
  • 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: simherm
    • SimilarityGroup(V): ModTupFld → GrpMat
  • Gram-Schmidt Normalisation
    • GramSchmidtPair(J): AlgMatElt → AlgMatElt, AlgMatElt
    • Example: hermitiangs
    • Example: skewgs
• Classical Groups
  • Example: fixaltform
  • Example: fixhermform
• Lie Algebras and Bilinear Forms
  • DerivationAlgebra(J): AlgMatElt → AlgLie
  • Example: lie C3
  • Example: lie G2
  • HeisenbergAlgebra(J): AlgMatElt → AlgLie
  • Example: heisenberg
• 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: reducible
  • Example: nonabs
  • InvariantQuadraticForms(G): GrpMat → SeqEnum[AlgMatElt]
  • Example: invquadform
  • SemilinearDual(M, mu): ModGrp, Map → ModGrp
  • InvariantSesquilinearForms(G): GrpMat → SeqEnum[AlgMatElt]
  • Example: sesquiforms
  • Example: hermandalt
  • 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: semiinv
• Polar Spaces More Generally
  • Creation of PolarSpaces
    • AmbientPolarSpace(J): AlgMatElt → SpcPlr
    • AmbientPolarSpace(J, a): AlgMatElt, FldAut → SpcPlr
    • PolarSpace(L): LatNF → SpcPlr
    • PolarSpace(L): Lat → SpcPlr
    • AmbientSpace(L): LatNF → SpcPlr
    • ChangeRing(V, R): SpcPlr, Rng → SpcPlr
  • Properties of Polar Spaces
    • BaseField(V): SpcPlr → Fld
    • BaseRing(V): SpcPlr → Fld
    • CoefficientField(V): SpcPlr → Fld
    • CoefficientRing(V): SpcPlr → Fld
    • VectorSpace(V): SpcPlr → ModTupFld
    • KSpace(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