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). The standard forms are preserved by the Magma implementation of the classical groups over finite fields.

If \(J\) is the matrix of a bilinear form, \(X\) preserves the form if \(XJX^\mathrm{tr}= J\). If \(J\) represents an hermitian form and if \(\overline{X}\) is obtained from \(X\) by application of the field automorphism, then \(X\) preserves the form if \(XJ\overline{X}^\mathrm{tr}= J\).

If \(\beta\) is a non-degenerate alternating form, then \(\hbox{rad}(V)\) 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 \(n\times n\) matrix of a non-degenerate alternating form over the ring \(R\) (or the field of \(q\) elements) such that if \(e_1\), \(e_2\), …, \(e_{2m}\) is the standard basis, then \((e_1,e_{2m})\), \((e_2,e_{2m-1})\), …, \((e_m,e_{m+1})\) are mutually orthogonal hyperbolic pairs.

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

Example: alternatingform (ex-3c60cd)

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

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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 \(n\times n\) anti-diagonal matrix \((\delta_{i,n-i+1})\) over the field \(K\) (or the field of \(q^2\) elements). If \(K\) is the finite field of \(q^2\) elements, the second return value is the field involution \(K \to K :x \mapsto x^q\). 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 \(\beta\) is a non-degenerate hermitian form over a finite field, then \(\hbox{rad}(V)\) 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 \(\hbox{GU}(n,q)\).

StandardQuadraticForm(n, K : parameters): RngIntElt, Fld -> AlgMatElt

StandardQuadraticForm(n, q : parameters): RngIntElt, RngIntElt -> AlgMatElt

Minus  : BoolElt                      Default: false
Variant: MonStgElt                    Default: "Default"

An \(n\times n\) 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 \(\delta_{i,n-i+1}\), where \(1 \le i \le (n+1)/2\).

If Minus is true and \(n = 2m\), this function returns a form whose Witt index is \(m-1\). If Minus is true, \(n = 2m+1\) and \(K\) is a finite field of odd characteristic, the return value is a form of Witt index \(m\) which is similar to but not congruent to the default form. Use the Revised option of Variant to obtain the form preserved by the orthogonal groups AltGOMinus(n,q) and subgroups.

If \(K\) is a finite field of order \(q\) and \(W\) is the anisotropic component of a hyperbolic splitting of a form of even rank and Minus type, then \(W\) has basis vectors \(e\) and \(f\) such that \(Q(e) = -1\), \(\beta(e,f) = -\xi-\xi^q\) and \(Q(f) = -\xi\xi^q\), where \(\xi\) is a primitive element of \({\bf F}_{q^2}\). If the characteristic of \(K\) is odd, the Revised option returns a form corresponding to an orthonormal basis for \(W\).

Example: minusform (ex-78f890)

Construct a standard quadratic form of minus type.

> K<z> := GF(7,2);
> Q := StandardQuadraticForm(4,49 : Minus);
> Q;
[   0    0     0    1]
[   0    6     5    0]
[   0    0  z^29    0]
[   0    0     0    0]
> _<x> := PolynomialRing(K);
> a := Q[2,2] * Q[3,3];
> IsIrreducible(x^2+x+a);
true

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Example: revisedminus (ex-29b8f5)

Compare the revised form with the standard form: the forms Q above and QR below have different entries in the central \(2\times 2\) 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]

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StandardSymmetricForm(n, K): RngIntElt, Fld -> AlgMatElt

StandardSymmetricForm(n, q : parameters): RngIntElt, RngIntElt -> AlgMatElt

Minus  : BoolElt                      Default: false
Variant: MonStgElt                    Default: "Default"

In all cases this is \(Q + Q^{\hbox{tr}}\), where \(Q\) is the corresponding standard quadratic form, as defined above.