Let β be a reflexive bilinear or a sesquilinear form on the vector space V. A non-zero vector v is isotropic (with respect to β) if β(v, v) = 0. If Q is a quadratic form, a non-zero vector v is singular if Q(v) = 0.
A polar space V has a hyperbolic splitting; namely, a direct sum decomposition
V = L1perp L2perp ... perp Lm perp W
where the Li are hyperbolic planes and m is maximal.
The polar space is hyperbolic if W = 0; i.e., it is an orthogonal sum of hyperbolic planes. In Bourbaki [Bou07, p. 66] the corresponding form is said to be neutral.
If the form defining the polar space is non-degenerate and not pseudo-alternating, then every isotropic (resp. singular) vector belongs to a hyperbolic pair. Therefore if the charactersistic is not 2, W is anisotropic; i.e., it does not contain any isotropic vectors. In this case the integer m is the Witt index of the form and W is called the anisotropic component of the splitting.
If the characteristic is 2 and V is a quadratic space, W does not contain singular vectors but it may contain isotropic vectors.
A non-degenerate polar space V of dimension 2m which is the direct sum of two totally isotropic subspaces is hyperbolic and it has a symplectic basis; i.e., a basis e1, f1, ..., em, fm such that the pairs (ei, fi), 1≤i≤m are mutually orthogonal hyperbolic pairs.
Determine whether the polar space V contains an isotropic vector; if it does, the second return value is a representative.
Determine whether the quadratic space V contains a singular vector; if it does, the second return value is a representative.
A subspace W of a polar space is totally isotropic if every non-zero vector of W is isotropic.
Returns true if the polar space V is totally isotropic, otherwise false.
A subspace W of a quadratic space defined by a quadratic form Q is totally singular if Q(w) = 0 for all w∈W.
Returns true if the quadratic space V is totally singular, otherwise false.
A representative maximal totally isotropic subspace of the polar space V.
A representative maximal totally singular subspace of the quadratic space V.
The Witt index of a polar space V that is not a quadratic space is the dimension of a maximal totally isotropic space. The Witt index of a quadratic space is the dimension of a maximal totally singular subspace.
If the characteristic of the field is not 2 and if β is the polar form of Q, a subspace is totally singular if and only if it is totally isotropic with respect to β; in this case the Witt index of Q coincides with the Witt index of β.
The Witt index of the polar space V.
An ordered pair of vectors (u, v) such that u and v are isotropic and β(u, v) = 1 is a hyperbolic pair. If V is a quadratic space, u and v are required to be singular. The subspace spanned by a hyperbolic pair is a hyperbolic plane.
If V is a pseudo-symplectic space defined by a symmetric bilinear form β over a finite field of characteristic 2, define the pseudo-radical of V to be the radical of the hyperplane { v∈V | β(v, v) = 0}.
Given a singular or isotropic vector u which is not in the radical or pseudo-radical, return a vector v such that (u, v) is a hyperbolic pair.
> V := VectorSpace(GF(2),2);
> IsPseudoSymplecticSpace(V);
true
> IsNondegenerate(V);
true
> { v : v in V | v ne V!0 and DotProduct(v,v) eq 0};
{
(1 1)
}
> Q := StandardQuadraticForm(6,4 : Minus); > V := QuadraticSpace(Q); > WittIndex(V); 2 > H := HyperbolicSplitting(V); > W := sub< V | H[2] >; > HasSingularVector(W); false > HasIsotropicVector(W); true ( 0 0 1 0 0 0)
A pair (M, B), where M is a maximal list of pairwise orthogonal hyperbolic pairs and B is a basis for the orthogonal complement of the subspace they span. This function requires the form to be non-degenerate and, except for symplectic spaces, the base ring of V must be a finite field.
> K<a> := GF(7,2);
> J := Matrix(K,3,3,[1,2,1, 2,1,0, 1,0,2]);
> V := VectorSpace(K,3,J);
> W := sub<V| [a,a,a], [1,2,3]>;
> IsNondegenerate(W);
true
> HyperbolicSplitting(W);
<[
[
(a^20 1 a^39),
(a^12 2 a)
]
], []>
> IsNondegenerate(V);
false
> R := Radical(V);
> H := (Dimension(R) eq 0) select V else
> sub<V|[e : e in ExtendBasis(B,V) | e notin B] where B is Basis(R)>;
> HyperbolicSplitting(H);
<[
[
( 0 a^20 1),
( 0 a^12 2)
]
], []>
Given totally isotropic subspaces U and W of a non-degnerate polar space V such that V = U direct-sum W, return a symplectic basis for V such that e1, e2, ..., em is a basis for U and f1, f2, ..., fm is a basis for W.
Let V = L1perp ... perp Lm perp W perp rad(V) be a hyperbolic splitting of the polar space V where the Li are hyperbolic planes spanned by hyperbolic pairs (ei, fi) for 1≤i≤m. The subspaces P = < e1, ..., em > and N = < f1, ..., fm > are totally isotropic (resp. totally singular) and we call the 4-tuple (rad(V), P, N, W) a Witt decomposition of V.
The Witt decomposition of the space V.
A quadratic space is metabolic if it is a direct sum E direct-sum F of totally singular subspaces E and F such that E = Eperp.
Given a quadratic space with quadratic form q : V to F, the metabolic space based on V is the quadratic space M = V direct-sum V * with quadratic form Q : M to F defined by Q(v, f) = q(v) + vf.
The metabolic space based on the quadratic space V.