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Isotropic and Singular Vectors and Subspaces

Let βbe 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 subspace W of a polar space V is totally isotropic if every non-zero vector of W is isotropic. If V is a quadratic space, W is totally singular if every non-zero vector of W is singular.

HasIsotropicVector(V) : ModTupFld -> BoolElt, ModTupFldElt
Determine whether the polar space V contains an isotropic vector; if it does, the second return value is a representative.
HasSingularVector(V) : ModTupFld -> BoolElt, ModTupFldElt
Determine whether the quadratic space V contains a singular vector; if it does, the second return value is a representative.

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.

HyperbolicPair(V, u) : ModTupFld, ModTupFldElt -> ModTupFldElt
Given a singular or isotropic vector u which is not in the radical, return a vector v such that (u, v) is a hyperbolic pair.

If V is the direct sum of subspaces U and W and if β(u, w) = 0 for all u∈U and all w∈W, we write V = U perp W.

A vector space V furnished with a reflexive form βhas a direct sum decomposition V = U perp rad(V), where U is any complement to rad(V) in V.

If V is a polar space, it has a hyperbolic splitting; namely, it is a direct sum

V = L1perp L2perp ... perp Lm perp W

where the Li are 2-dimensional subspaces spanned by hyperbolic pairs and m is maximal. 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 and consequently W does not contain any isotropic (resp. singular) vectors. In this case the integer m is the Witt index of the form and W is called the anisotropic component of the splitting. A non-degenerate form on V is said to have maximal Witt index if dim V is 2m or 2m + 1.


Example FldForms_pseudoalt (H29E4)

The vector space of dimension 2 over GF(2) is pseudo-symplectic (the form is the identity matrix). It has three non-zero elements only one of which is isotropic. This confirms that not every isotropic vector in a non-degenerate pseudo-symplectic space belongs to 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)
}

HyperbolicSplitting(V) : ModTupFld -> SeqEnum, SeqEnum
A maximal list of pairwise orthogonal hyperbolic pairs together with 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.

Example FldForms_hypsplit (H29E5)

Find the hyperbolic splitting of a polar space defined by a symmetric bilinear form. In this example W is a non-degenerate subspace of the polar space V.

> 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)
    ]
], []>

Example FldForms_extradical (H29E6)

The polar space V of the previous example is degenerate and so HyperbolicSplitting cannot be applied directly. Instead, we first split off the radical.

> 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)
    ]
], []>


IsTotallyIsotropic(V) : ModTupFld) -> BoolElt
Returns true if the polar space V is totally isotropic, otherwise false.
IsTotallySingular(V) : ModTupFld) -> BoolElt
Returns true if the quadratic space V is totally singular, otherwise false.

Suppose that V = L1perp ... perp Lm perp W perp rad(V) where the Li are 2-dimensional subspaces spanned by hyperbolic pairs (ei, fi) for 1≤i≤m. The subspaces P = < e1, ..., em > and N = < f1, ..., fm > are totally isotropic and we call the 4-tuple (rad(V), P, N, W) a Witt decomposition of V. A polar space is hyperbolic if it is the direct sum of two totally isotropic (resp. totally singular) subspaces; in Bourbaki [Bou07, p. 66] the corresponding form is said to be neutral.

WittDecomposition(V) : ModTupFld -> SeqEnum[ModTupFld]
The Witt decomposition of the space V.
WittIndex(V) : ModTupFld -> RngIntElt
The Witt index of the polar space V; namely half the dimension of a maximal hyperbolic subspace.
MaximalTotallyIsotropicSubspace(V) : ModTupFld -> ModTupFld
A representative maximal totally isotropic subspace of the polar space V.
MaximalTotallySingularSubspace(V) : ModTupFld -> ModTupFld
A representative maximal totally singular subspace of the quadratic space V.
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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013