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Constructing Polar Spaces

If J is an n x n matrix, the command VectorSpace(K,n,J) creates a vector space of dimension n over K with a bilinear form whose matrix is J.

The default form attached to every vector space in Magma is the symmetric form whose matrix is the identity matrix.

A vector space V is recognised as a polar space if any of the following conditions apply. (There is no check to ensure that the inner product matrix is is non-degenerate.)

1.
There is a quadratic form attached to V.
2.
There is a field involution attached to V and the inner product matrix of V is hermitian or skew-hermitian with respect to this involution.
3.
The inner product matrix of V is symmetric or alternating.

Thus a vector space with a symmetric inner product matrix but no quadratic form attached is a polar space. If the characteristic of the field is 2 and the form is not alternating it is a pseudo-symplectic space, otherwise we shall call it an orthogonal space to distinguish it from quadratic spaces.
Subsections
IsPolarSpace(V) : ModTupFld -> BoolElt
Check if the vector space V is a quadratic space or if the Gram matrix of V is a reflexive form.
PolarSpaceType(V) : ModTupFld -> MonStgElt
The type of the polar space V, returned as a string.

Example FldForms_polarspace (H29E10)

Create the standard vector space of dimension 4 over the rational field and check if it is a polar space.

> V := VectorSpace(Rationals(),4);
> IsPolarSpace(V);
true
> PolarSpaceType(V);
orthogonal space


Symplectic Spaces

SymplecticSpace(J) : AlgMatElt -> ModTupRng
The symplectic space of dimension n defined by the n x n matrix J. This function checks to ensure that J is alternating.
IsSymplecticSpace(W) : ModTupFld -> BoolElt
Returns true if the Involution attribute of the generic vector space W is not assigned and the space carries an alternating form, otherwise false.

Note that a quadratic space over a field of characteristic 2 satisfies these conditions and consequently this function will return true for these spaces.

IsPseudoSymplecticSpace(W) : ModTupFld -> BoolElt
Given a vector space W over a finite field, this intrinsic returns true if the base field has characteristic 2, the Involution attribute is not assigned to the generic space and the form is symmetric but not alternating, otherwise false.

Unitary Spaces

In order to accommodate hermitian forms it is necessary to assign a field automorphism of order two to the Involution attribute of the vector space.

Thus a unitary space is characterised as a vector space V whose ambient space, Generic(V), has the attribute Involution and whose inner product matrix is either hermitian or skew hermitian.

UnitarySpace(J, sigma) : AlgMatElt, Map -> ModTupFld
The n-dimensional unitary space over the base field K of J, where σis an automorphism of K of order 2 and where J is an n x n matrix which is hermitian or skew-hermitian with respect to σ.
IsUnitarySpace(W) : ModTupFld -> BoolElt
Return true if the Involution attribute of the generic space of W is assigned and the form is either hermitian or skew-hermitian when restricted to W.

Example FldForms_unitaryform (H29E11)

Create a unitary geometry with the standard hermitian form and check that the given vector is isotropic. Note that the function DotProduct takes both the form and the field involution into account when calculating its values. For finite fields, the function InnerProduct ignores the field involution.

> K<z> := GF(25);
> J, sigma := StandardHermitianForm(5,K);
> J;
[   0    0    0    0    1]
[   0    0    0    1    0]
[   0    0    1    0    0]
[   0    1    0    0    0]
[   1    0    0    0    0]
> sigma(z);
z^5
> V := UnitarySpace(J,sigma);
> u := V![1,z,0,z^2,-1];
> DotProduct(u,u);
0
> InnerProduct(u,u);
z^20


ConjugateTranspose(M, sigma) : Mtrx, Map -> Mtrx
The transpose of the matrix σ(M), where σis an automorphism of the base field ofthe matrix M.

Quadratic Spaces

A vector space V with an attached quadratic form is called a quadratic space. The polar form of a quadratic space is the inner product matrix J of the space. If the characteristic of the field is not 2, the value of the quadratic form on a row vector v is (1/2)(v * J * vtr).

QuadraticSpace(Q) : AlgMatElt -> ModTupFld
The quadratic space of dimension n defined by the quadratic form represented by an upper triangular n x n matrix Q. The inner product matrix of the space is Q + Qtr. If Q is not upper triangular, the space will be constructed but there is no guarantee that other functions in this package will return correct results.
QuadraticSpace(f) : RngMPolElt -> ModTupFld
The quadratic space of dimension n whose quadratic form is given by the quadratic polynomial f in n variables. If the variables are x1, ..., xn, then for i≤j, the (i, j)-th entry of the matrix of the form is the coefficient of xixj in f.
SymmetricToQuadraticForm(J) : AlgMatElt -> AlgMatElt
Provided the characteristic of the field is not two, this is the upper triangular matrix which represents the same quadratic form as the symmetric matrix J.
QuadraticFormMatrix(V) : ModTupRng -> ModAlgElt
The (upper triangular) matrix which represents the quadratic form of the quadratic space V.
QuadraticNorm(v) : ModTupFldElt -> FldElt
The value Q(v), where Q is the quadratic form attached to the generic space of the parent of the vector v.
QuadraticFormPolynomial(V) : ModTupRng -> RngPolElt
The polynomial ∑i≤j qijxixj, where Q = (qij) is the matrix of the quadratic form of the quadratic space V.

Example FldForms_polyquad (H29E12)

The quadratic space defined by a polynomial.

> _<x,y,z> := PolynomialRing(Rationals(),3);
> f := x^2 + x*y +3*x*z - 2*y*z + y^2 +z^2;
> V := QuadraticSpace(f);
> PolarSpaceType(V);
quadratic space
> IsNonsingular(V);
true
> QuadraticFormMatrix(V);
[ 1  1  3]
[ 0  1 -2]
[ 0  0  1]


OrthogonalSum(V, W) : ModTupFld, ModTupFld) -> ModTupFld
The orthogonal direct sum of the quadratic spaces V and W.
TotallySingularComplement(V, U, W) : ModTupFld, ModTupFld, ModTupFld) -> ModTupFld
Given totally singular subspaces U and W of the quadratic space V such that Uperp ∩W = 0 this function returns a totally singular subspace X such that V = X direct-sum Uperp and W⊆X.
Discriminant(V) : ModTupFld -> RngIntElt
If V is a vector space over the finite field K and if J is the Gram matrix of V, the discriminant of V is the determinant Δof J modulo squares. That is, the discriminant is 0 if Δis a square in K, 1 if it is a non-square. The form J is required to be non-degenerate.
ArfInvariant(V) : ModTupFld -> RngIntElt
The Arf invariant of the quadratic space V.

Currently this is available only for quadratic spaces of even dimension 2m over a finite field F of characteristic 2. In this case there are two possibilities: either the Witt index of the form is m and the Arf invariant is 0, or the Witt index is m - 1 and the Arf invariant is 1.

DicksonInvariant(V, f) : ModTupFld, Mtrx -> RngIntElt
The Dickson invariant of the isometry f of the quadratic space V is the rank (mod 2) of 1 - f. If the polar form of Q is non-degenerate, the Dickson invariant defines a homomorphism from the orthogonal group O(V) onto the additive group of order 2.
SpinorNorm(V, f) : ModTupFld, Mtrx -> RngIntElt
The spinor norm of the isometry f of the quadratic space V. This is the discriminant of the Wall form (Section Wall Forms) of f.
HyperbolicBasis(U, B, W) : ModTupFld, SeqEnum, ModTupFld -> SeqEnum
Given complementary totally singular subspaces U and W of a quadratic space and a basis B for U, return a sequence of pairwise orthogonal hyperbolic pairs whose second components form a basis for W.
OrthogonalReflection(a) : ModTupFldElt -> AlgMatElt
The reflection determined by a non-singular vector of a quadratic space.
RootSequence(V, f) : ModTupFld, Mtrx -> SeqEnum
Given a matrix f representing an isometry of the quadratic space V, return a sequence of vectors such that the product of the corresponding orthogonal reflections is f. The empty sequence is returned if f is the identity matrix.
ReflectionFactors(V, f) : ModTupFld, Mtrx) -> SeqEnum
Given a matrix f representing an isometry of the quadratic space V, return a sequence of reflections whose product is f. The empty sequence corresponds to the identity matrix.

Given a quadratic space V defined by a quadratic form Q with polar form βand non-zero vectors u, v ∈V such that u is singular and β(u, v) = 0, the Siegel transformation (also called an Eichler transformation) is the isometry ρu, v defined by

x ρu, v = x + β(x, v)u - β(x, u)v - Q(v)β(x, u) u.

SiegelTransformation(u, v) : ModTupFldElt, ModTupFldElt -> AlgMatElt
The Siegel transformation defined by a singular vector u and a vector v orthogonal to u. The common parent of u and v must be a quadratic space.

Example FldForms_siegel (H29E13)

A group of isometries generated by Siegel transformations.

> Q := StandardQuadraticForm(4,3);
> V := QuadraticSpace(Q);
> u := V.1;
> QuadraticNorm(u);
0
> X := { v : v in V | DotProduct(u,v) eq 0 and QuadraticNorm(v) ne 0 };
> #X;
12
> H := sub< GL(V) | [SiegelTransformation(u,v) : v in X]>;
> #H;
9


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Version: V2.19 of Mon Dec 17 14:40:36 EST 2012