Polar Spaces
- Introduction
- Reflexive Forms
- Inner Products
- Isotropic and Singular Vectors and Subspaces
HasIsotropicVector(V): ModTupFld → BoolElt, ModTupFldEltHasSingularVector(V): ModTupFld → BoolElt, ModTupFldEltIsTotallyIsotropic(V): ModTupFld → BoolEltIsTotallySingular(V): ModTupFld → BoolEltMaximalTotallyIsotropicSubspace(V): ModTupFld → ModTupFldMaximalTotallySingularSubspace(V): ModTupFld → ModTupFldWittIndex(V): ModTupFld → RngIntEltHyperbolicPair(V, u): ModTupFld, ModTupFldElt → ModTupFldElt- Example: Isotropic Vectors in a Pseudo-Symplectic Space
- Example: Hyperbolic Splitting in Characteristic 2
HyperbolicSplitting(V): ModTupFld → Tup- Example: Hyperbolic Splitting of a Symmetric Form
- Example: Splitting Off the Radical
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 → AlgMatEltStandardAlternatingForm(n,q): RngIntElt, RngIntElt → AlgMatElt- Example: Standard Alternating Form
StandardPseudoAlternatingForm(n,K): RngIntElt, Fld → AlgMatEltStandardPseudoAlternatingForm(n,q): RngIntElt, RngIntElt → AlgMatEltStandardHermitianForm(n,K): RngIntElt, Fld → AlgMatElt, MapStandardHermitianForm(n,q): RngIntElt, RngIntElt → AlgMatElt, MapStandardQuadraticForm(n, K : parameters): RngIntElt, Fld → AlgMatEltStandardQuadraticForm(n, q : parameters): RngIntElt, RngIntElt → AlgMatElt- Example: Standard Quadratic Form of Minus Type
- Example: Revised Minus-Type Quadratic Form
StandardSymmetricForm(n, K): RngIntElt, Fld → AlgMatEltStandardSymmetricForm(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 → BoolEltPolarSpaceType(V): ModTupFld → MonStgElt- Example: Testing for a Polar Space
- Isotropic Subspaces
- Symplectic Spaces
- Unitary Spaces
- Quadratic Spaces
QuadraticSpace(Q): AlgMatElt → ModTupRngQuadraticSpace(f): RngMPolElt → ModTupRngSymmetricToQuadraticForm(J): AlgMatElt → AlgMatEltQuadraticFormMatrix(V): ModTupRng → ModAlgEltQuadraticNorm(v): ModTupFldElt → FldEltQuadraticFormPolynomial(V): ModTupRng → RngPolEltQuadraticFormPolynomial(Q): AlgMatElt → RngPolElt- Example: Quadratic Space from a Polynomial
OrthogonalSum(V, W): ModTupFld, ModTupFld → ModTupFld, Map, MapOrthogonalTensorProduct(V, W): ModTupFld, ModTupFld → ModTupFldTotallySingularComplement(V, U, W): ModTupFld, ModTupFld, ModTupFld → ModTupFldDiscriminant(V): ModTupFld → RngIntEltArfInvariant(V): ModTupFld → RngIntEltDicksonInvariant(V, f): ModTupFld, Mtrx → RngIntEltSpinorNorm(V, f): ModTupFld, Mtrx → RngIntEltSpinorNorm(Q, g): AlgMatElt[Fld], AlgMatElt[Fld] → FldElt- Example: Spinor Norm over a General Field
CharacterQQModSquares(d, r): RngIntElt, FldRatElt → RngIntEltSpinorNormRho(d, g, Q): RngIntElt, GrpMatElt, AlgMatElt → RngIntEltHyperbolicBasis(U, B, W): ModTupFld, SeqEnum, ModTupFld → SeqEnumOrthogonalReflection(a): ModTupFldElt → AlgMatEltRootSequence(V, f): ModTupFld, Mtrx → SeqEnumReflectionFactors(V, f): ModTupFld, Mtrx → SeqEnumSiegelTransformation(u, v): ModTupFldElt, ModTupFldElt → AlgMatElt- Example: Isometries from Siegel Transformations
- Isometries and Similarities
- Isometries
IsIsometry(U, V, f): ModTupFld, ModTupFld, Map → BoolEltIsIsometry(f): Map → BoolEltIsIsometry(V, g): ModTupFld, Mtrx → BoolEltIsIsometric(V, W): ModTupFld, ModTupFld → BoolElt, Map- Example: Isometric Polar Spaces
- Example: Isometry Between Symplectic Spaces
- Example: Transforming Forms with TransformForm
CommonComplement(V, U, W): ModTupFld, ModTupFld, ModTupFld → ModTupFldExtendIsometry(V, U, f): ModTupFld, ModTupFld, Map → MapIsometryGroup(V): ModTupFld → GrpMat- Example: Isometry Group of a Degenerate Quadratic Space
- Example: Conjugating Isometry Groups
- Similarities
IsSimilarity(U, V, f): ModTupFld, ModTupFld, Map → BoolElt, FldEltIsSimilarity(f): Map → BoolElt, FldEltIsSimilarity(V, g): ModTupFld, Mtrx → BoolElt, FldEltIsSimilar(V, W): ModTupFld, ModTupFld → BoolElt, Map- Example: Similar Hermitian and Skew-Hermitian Spaces
SimilarityGroup(V): ModTupFld → GrpMat
- Gram-Schmidt Normalisation
- Classical Groups
- Lie Algebras and Bilinear Forms
- Wall Forms
WallForm(V, f): ModTupFld, Mtrx → ModTupFld, MapWallIsometry(V, I, mu): ModTupFld, ModTupFld, Map → MtrxWallDecomposition(V, f): ModTupFld, Mtrx → Mtrx, MtrxSemiOrthogonalBasis(V): ModTupFld → SeqEnumGeneralisedWallForm(V, f): ModTupFld, Mtrx → ModTupFld, Map
- Invariant Forms
- Polar Spaces More Generally
- Bibliography