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The Standard Forms

This section describes the "standard" alternating, hermitian, quadratic and symmetric forms defined on a finite dimensional vector space over a field. These are forms of maximal Witt index together with the quadratic forms of non-maximal Witt index over finite fields (see Section Isotropic and Singular Vectors and Subspaces). Except for the orthogonal groups the standard forms are preserved by the Magma implementation of the classical groups over finite fields.

If J is the matrix of the form, then X preserves the form if XJXtr = J.

If βis a non-degenerate alternating form, then rad(V) and the anisotropic component of a hyperbolic splitting are zero. Thus the dimension of V must be even and V has a basis of mutually orthogonal hyperbolic pairs. In particular, up to equivalence, there is only one non-degenerate alternating form on V.

StandardAlternatingForm(n,R) : RngIntElt, Rng -> AlgMatElt
StandardAlternatingForm(n,q) : RngIntElt, RngIntElt -> AlgMatElt
If n = 2m, this function returns the n x n matrix of a non-degenerate alternating form over the ring R (or the field of q elements) such that if e1, e2, ..., e2m is the standard basis, then (e1, e2m), (e2, e2m - 1), ..., (em, em + 1) are mutually orthogonal hyperbolic pairs.

The group of isometries of this form is the symplectic group Sp(2m, R).


Example FldForms_alternatingform (H29E7)

Create a symplectic geometry with the standard alternating form and then check that every non-zero vector is isotropic.

> K := GF(5);
> J := StandardAlternatingForm(4,K);
> J;
[0 0 0 1]
[0 0 1 0]
[0 4 0 0]
[4 0 0 0]
> V := VectorSpace(K,4,J);
> forall{ v : v in V | DotProduct(v,v) eq 0 };
true

StandardPseudoAlternatingForm(n,K) : RngIntElt, Fld -> AlgMatElt
StandardPseudoAlternatingForm(n,q) : RngIntElt, RngIntElt -> AlgMatElt
The matrix of the standard pseudo-alternating form of degree n over the field K (or the finite field of order q), which must have characteristic 2; that is, a symmetric form which is not alternating.
StandardHermitianForm(n,K) : RngIntElt, Fld -> AlgMatElt, Map
StandardHermitianForm(n,q) : RngIntElt, RngIntElt -> AlgMatElt, Map
The first return value of this function is the n x n anti-diagonal matrix (δi, n - i + 1) over the field K (or the field of q2 elements). If K is the finite field of q2 elements, the second return value is the field involution K to K :x |-> xq. If K is a field which admits the operation of complex conjugation, the second return value is the field automorphism which sends each element to its complex conjugate.

If βis a non-degenerate hermitian form over a finite field, then rad(V) is zero and the dimension of the anisotropic component of a hyperbolic splitting is either 1 or 0.

In the finite field case, the group of isometries of this form is GU(n, q).

Suppose that βis the polar form of a quadratic form Q defined on the vector space V. A vector v∈V is said to be singular if Q(v) = 0. A subspace W is totally singular is Q(w) = 0 for all w∈W. The Witt index of Q is the dimension of a maximal totally singular subspace. If the characteristic of the field is not 2 a subspace is totally singular if and only if it is totally isotropic with respect to βand hence in this case the Witt index of Q coincides with the Witt index of β.

StandardQuadraticForm(n, K : parameters) : RngIntElt, Fld -> AlgMatElt
StandardQuadraticForm(n, q : parameters) : RngIntElt, RngIntElt -> AlgMatElt
    Minus: BoolElt                      Default: false
    Variant: MonStgElt                  Default: "Default"
An n x n upper triangular matrix representing a quadratic form over the field K (or the field of order q). The default option is to return a form of maximal Witt index, namely the upper triangular matrix whose non-zero entries are δi, n - i + 1, where 1 ≤i ≤(n + 1)/2.

If Minus is true and n = 2m, this function returns a form whose Witt index is m - 1. This option is available only for finite fields.

For historical reasons, over finite fields of odd characteristic, the (m, m) element in the quadratic form used by the orthogonal groups of odd degree is 1/4.

If the K is a finite field and W is the anisotropic component of a hyperbolic splitting of a form of Minus type, then W has basis vectors e and f such that Q(e) = β(e, f) = 1 and Q(f) = a, where x2 + x + a is an irreducible polynomial in F[x]. This is the form returned by the Default option. If the characteristic of K is two, this is also returned by the Revised option. However, if the characteristic of K is odd, the Revised option returns a form corresponding to an orthonormal basis for W. The Original option returns the form preserved by OldGOMinus(2*m,q).


Example FldForms_minusform (H29E8)

Construct a standard quadratic form of minus type.

> K<z> := GF(7,2);
> Q := StandardQuadraticForm(4,49 : Minus);
> Q;
[   0    0     0    1]
[   0    1     1    0]
[   0    0  z^23    0]
[   0    0     0    0]
> _<x> := PolynomialRing(K);
> a := Q[3,3]; 
> IsIrreducible(x^2+x+a);
true

Example FldForms_revisedminus (H29E9)

Compare the revised form with the standard form: the forms Q above and QR below have different entries in the central 2 x 2 block.

> QR := StandardQuadraticForm(4,49 : Minus, Variant := "Revised");
> QR;
[   0    0     0    1]
[   0    4     0    0]
[   0    0  z^11    0]
[   0    0     0    0]

StandardSymmetricForm(n, K) : RngIntElt, Fld -> AlgMatElt
StandardSymmetricForm(n, q : parameters) : RngIntElt, RngIntElt -> AlgMatElt
    Minus: BoolElt                      Default: false
    Variant: MonStgElt                  Default: "Default"

In all casses this is Q + Q^(tr), where Q is the corresponding standard quadratic form, as defined above.

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