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.
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.
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.
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.
> 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) }
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.
> 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) ] ], []>
Returns true if the polar space V is totally isotropic, otherwise false.
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.
The Witt decomposition of the space V.
The Witt index of the polar space V; namely half the dimension of a maximal hyperbolic subspace.
A representative maximal totally isotropic subspace of the polar space V.
A representative maximal totally singular subspace of the quadratic space V.[Next][Prev] [Right] [Left] [Up] [Index] [Root]