Morphisms of forms

In this chapter we give a very brief overview on morphisms of sesquilinear and quadratic forms. The reader can find more in the texts: Cameron , Taylor , Aschbacher , or Kleidman and Liebeck .

In this chapter we consider an n-dimensional vector space V over a finite field. Suppose that f is a sesquilinear form or a quadratic form on V, then we call the pair (V,f) a formed vector space.

Morphisms of sesquilinear forms
Consider two formed vector spaces (V,f) and (W,g) over the same field F, where both f and g are sesquilinear forms. Suppose that \phiφ is a linear map from V to W.

The map \phiφ is an isometry Isometry from the formed space (V,f) to the formed space (W,g) if for all v,w in V we have f(v,w) = f'(\phi(v), \phi(w)). f(v,w) = f'(φ(v),φ(w)) The map \phiφ is a similarity Similarity from the formed space (V,f) to the formed space (W,g) if for all v,w in V we have f(v,w) = \lambda f'(\phi(v), \phi(w)). f(v,w) = λ (f'(φ(v),φ(w))) for some non-zero λ in F\lambda \in F. Finally, the map φ\phi. is a semi-similarity Semi-similarity from the formed space (V,f) to the formed space (W,g) if for all v,w in V we have f(v,w) = \lambda f'(\phi(v), \phi(w))^\alpha f(v,w) = λ f'(φ(v),φ(w))<SUP>α</SUP> for some non-zero λ in F\lambda \in F and a field automorphism α\alpha of F.

One of the objectives of studying maps between formed vector spaces is the classification of sesquilinear forms on a vector space V, where it is sufficient to classify non-degenerate forms. The following results are well known.

It can be proved that (see for example Section 6.3 of ): all non-degenerate alternating forms of a given vector space over a given finite field are similar, all non-degenerate hermitian forms of a given vector space over a given finite field are similar, and, the non-degenerate symmetric bilinear forms on a vector space over a field with odd characteristic come in three flavours, two of which occur when the dimension of the vector space is even, one of which occurs when the dimension of the vector space is odd. In principle, within each similarity class, different isometry classes can occur, but we will see that in most cases, each similarity class contains exactly one isometry class.

Given a sesquilinear form f over a vector space V, `Forms` provides functionality to compute the linear map \phiφ from V to itself (or, equivalently, a matrix describing a change of basis), such that f is mapped to its canonical representative in its isometry class. In the next sections, we describe the representative(s) of the similarity class(es) used in `Forms`, and, when necessary, the different isometry classes, for each of the three reflexive sesquilinear forms. The easiest cases are the hermitian and alternating cases.

Hermitian forms
We suppose that f is a non-degenerate hermitian form on a vector space V over the finite field F, with involutory field automorphism \alphaα. It can be proved (see ) that any vector space equipped with a non-degenerate hermitian form f contains an orthogonal basis such that f(e_i,e_i)=1f(e<SUB>i</SUB>,e<SUB>j</SUB>)=1 for each basisvector e_ie<SUB>i</SUB>. Hence (V,f) is isometric with (V,f') with f' the non-degenerate hermitian form with the identity matrix over F. The Witt index of f equals n/2 when n is even and (n-1)/2 when n is odd.
Alternating forms
We suppose that f is a non-degenerate alternating bilinear form on a vector space V over a finite field F. As already mentioned in Section , non-degenerate alternating forms only exist on even dimensional vector spaces. Restricting to a two dimensional vector space, it is clear immediately that the Gram matrix of f must be <!-- matrix expression begin --> <table style="color:#000"><tr> <td><table border=0 cellpadding=0 cellspacing=0px style="border-left:1px solid #000; border-right:1px solid #000; color:#000"><tr> <td style ="border-top:1px solid #000; border-bottom:1px solid #000;"> </td><td><table border=0 cellpadding=0 cellspacing=0 style="color:#000;"> <tr> <td align="center" valign="center" width=30>0</td> <td align="center" valign="center" width=30>r</td> </tr> <tr> <td align="center" valign="center" width=30>-r</td> <td align="center" valign="center" width=30>0</td> </tr> </table></td><td style ="border-top:1px solid #000; border-bottom:1px solid #000;"> </td></tr></table></td> </tr></table> <!-- matrix expression end --> \left( \begin{array}{cc}0 & r \\ -r & 0 \end{arry} \right) for some non-zero r \in F. If we rescale one of the basisvectors, which induces an isometry, then we see that there always exists a basis such that r=1. We call a two dimensional vector space equipped with a non-degenerate alternating form a symplectic hyperbolic line, and it is proved (see Theorem 6.7 of ) that the formed space (V,f) can be written as an orthogonal direct sum of symplectic hyperbolic planes. Hence, up to isometry, there is only one non-degenerate alternating form of an even dimensional vector space, and we choose as canonical representative the alternating form with Gram matrix \left( \begin{array}{ccccccc} 0 & 1 & 0 & 0 & \ldots & 0 & 0 \\ -1 & 0 & 0 & 0 & \ldots & 0 & 0 \\ 0 & 0 & 0 & 1 & \ldots & 0 & 0 \\ 0 & 0 & -1 & 0 & \ldots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \ldots & 0 & 1 \\ 0 & 0 & 0 & 0 & \ldots & -1 & 0 \\ \end{array} \right). <!-- matrix expression begin --> <table style="color:#000"><tr> <td><table border=0 cellpadding=0 cellspacing=0px style="border-left:1px solid #000; border-right:1px solid #000; color:#000"><tr> <td style ="border-top:1px solid #000; border-bottom:1px solid #000;"> </td><td><table border=0 cellpadding=0 cellspacing=0 style="color:#000;"> <tr> <td align="center" valign="center" width=30>0</td> <td align="center" valign="center" width=30>1</td> <td align="center" valign="center" width=30>0</td> <td align="center" valign="center" width=30>0</td> <td align="center" valign="center" width=30>...</td> <td align="center" valign="center" width=30>0</td> <td align="center" valign="center" width=30>0</td> </tr> <tr> <td align="center" valign="center" width=30>-1</td> <td align="center" valign="center" width=30>0</td> <td align="center" valign="center" width=30>0</td> <td align="center" valign="center" width=30>0</td> <td align="center" valign="center" width=30>...</td> <td align="center" valign="center" width=30>0</td> <td align="center" valign="center" width=30>0</td> </tr> <tr> <td align="center" valign="center" width=30>0</td> <td align="center" valign="center" width=30>0</td> <td align="center" valign="center" width=30>0</td> <td align="center" valign="center" width=30>1</td> <td align="center" valign="center" width=30>...</td> <td align="center" valign="center" width=30>0</td> <td align="center" valign="center" width=30>0</td> </tr> <tr> <td align="center" valign="center" width=30>0</td> <td align="center" valign="center" width=30>0</td> <td align="center" valign="center" width=30>-1</td> <td align="center" valign="center" width=30>0</td> <td align="center" valign="center" width=30>...</td> <td align="center" valign="center" width=30>0</td> <td align="center" valign="center" width=30>0</td> </tr> <tr> <td align="center" valign="center" width=30>:</td> <td align="center" valign="center" width=30>:</td> <td align="center" valign="center" width=30>:</td> <td align="center" valign="center" width=30>:</td> <td align="center" valign="center" width=30><sup><big>·</big></sup>·<sub><big>·</big></sub></td> <td align="center" valign="center" width=30>:</td> <td align="center" valign="center" width=30>:</td> </tr> <tr> <td align="center" valign="center" width=30>0</td> <td align="center" valign="center" width=30>0</td> <td align="center" valign="center" width=30>0</td> <td align="center" valign="center" width=30>0</td> <td align="center" valign="center" width=30>...</td> <td align="center" valign="center" width=30>0</td> <td align="center" valign="center" width=30>1</td> </tr> <tr> <td align="center" valign="center" width=30>0</td> <td align="center" valign="center" width=30>0</td> <td align="center" valign="center" width=30>0</td> <td align="center" valign="center" width=30>0</td> <td align="center" valign="center" width=30>...</td> <td align="center" valign="center" width=30>-1</td> <td align="center" valign="center" width=30>0</td> </tr> </table></td><td style ="border-top:1px solid #000; border-bottom:1px solid #000;"> </td></tr></table></td> </tr></table> The Witt index of f equals n/2.
Bilinear forms
We suppose that f is a non-degenerate symmetric bilinear form on a vector space V over a finite field F with odd characteristic. We call a two dimensional vector space a hyperbolic line if it contains a non-zero vector such that f(v,v) = 0. It is proved (see Proposition 6.9 of ) that any two hyperbolic lines are isometric, and we choose as canonical representative the orthogonal form with Gram matrix <!-- matrix expression begin --> <table style="color:#000"><tr> <td><table border=0 cellpadding=0 cellspacing=0px style="border-left:1px solid #000; border-right:1px solid #000; color:#000"><tr> <td style ="border-top:1px solid #000; border-bottom:1px solid #000;"> </td><td><table border=0 cellpadding=0 cellspacing=0 style="color:#000;"> <tr> <td align="center" valign="center" width=30>0</td> <td align="center" valign="center" width=30>1</td> </tr> <tr> <td align="center" valign="center" width=30>1</td> <td align="center" valign="center" width=30>0</td> </tr> </table></td><td style ="border-top:1px solid #000; border-bottom:1px solid #000;"> </td></tr></table></td> </tr></table> <!-- matrix expression end --> \left( \begin{array}{cc}0 & 1 \\ 1 & 0 \end{arry} \right).

It can be proved (see Theorem 6.10 of ) that the formed space (V,f) can be written as the orthogonal direct sum of hyperbolic lines and one subspace U of dimension at most two. The behaviour of f on the subspace U determines the similarity class of f. We describe the three occurring cases, to describe the chosen canonical form, we use the polynomial rather than the Gram matrix.

If the dimension of U is zero, then (V,f) is the orthogonal direct sum of hyperbolic lines, and hence (V,f) is isometric to the formed space (V,f'), where the Gram matrix of f' consists of blocks as described above. The chosen canonical form has polynomial x_1 x_2 + \ldots + x_{n-1}x_n x<sub>1</sub> x<sub>2</sub> + ... + x<sub>n<font face="symbol">-</font >1</sub>x<sub>n</sub> Note that the dimension of the vector space V is necessarily even. We call f hyperbolic (see also Section ). It follows also that in this similarity class, there is only one isometry class. The Witt index of f equals n/2.
If the dimension of U is one, then necessarily the polynomial of f equals \mu x_1^2 + x_2 x_3 + \ldots + x_{n-1}x_n μ x<sub>1</sub><sup>2</sup> + x<sub>2</sub> x<sub>3</sub> + ... + x<sub>n<font face="symbol">-</font >1</sub>x<sub>n</sub> for some \mu \in Fμ in F, and the dimension of the vector space V is odd. We call f parabolic (see also Section ). It is clear that if \muμ is a square in F, then rescaling the first basis vector yields a polynomial x_1^2 + x_2 x_3 + \ldots + x_{n-1}x_n x<sub>1</sub><sup>2</sup> + x<sub>2</sub> x<sub>3</sub> + ... + x<sub>n<font face="symbol">-</font >1</sub>x<sub>n</sub> which we choose as the canonical form for this similarity class. If \muμ is a non-square, a rescaling of x_2,x_4,\ldots,x_{n-1}x<sub>2</sub>,x<sub>4</sub>,...,x<sub>n<font face="symbol">-</font >1</sub> yields a polynomial \mu (x_1^2 + x_2 x_3 + \ldots + x_{n-1}x_n) μ(x<sub>1</sub><sup>2</sup> + x<sub>2</sub> x<sub>3</sub> + ... + x<sub>n<font face="symbol">-</font >1</sub>x<sub>n</sub>) which represents now a bilinear form that is similar but not isometric to the given one. Hence, the parabolic similarity class contains two isometry classes. The Witt index of f equals (n-1)/2.
Suppose at last that the dimension of U is two. We may suppose that U is not a hyperbolic line. It is not too difficult to see that a suitable base change yields the polynomial \mu x_1^2 + x_2^2 + x_3 x_4 + \ldots + x_{n-1}x_n μx<sub>1</sub><sup>2</sup> + x<sub>2</sub><sup>2</sup> + x<sub>3</sub> x<sub>4</sub> + ... + x<sub>n<font face="symbol">-</font>1</sub>x<sub>n</sub> for a non-square \mu \in Fμ in F, and the dimension of the vector space V is even. We call f elliptic. The Witt index of f equals (n-2)/2.
Degenerate forms
Suppose that f is a degenerate sesquilinear form on the vector space V, then Rad(f) is a non-trivial subspace of the vector space V. The vector space V can be written as the orthogonal direct sum of a subspace W and Rad(f), and the restriction of f to W is a non-degenerate sesquilinear form on W. Hence, f is isometric with a sesquilinear form having Gram matrix <!-- matrix expression begin --> <table style="color:#000"><tr> <td><table border=0 cellpadding=0 cellspacing=0px style="border-left:1px solid #000; border-right:1px solid #000; color:#000"><tr> <td style ="border-top:1px solid #000; border-bottom:1px solid #000;"> </td><td><table border=0 cellpadding=0 cellspacing=0 style="color:#000;"> <tr> <td align="center" valign="center" width=30>M</td> <td align="center" valign="center" width=30>A</td> </tr> <tr> <td align="center" valign="center" width=30>B</td> <td align="center" valign="center" width=30>C</td> </tr> </table></td><td style ="border-top:1px solid #000; border-bottom:1px solid #000;"> </td></tr></table></td> </tr></table> <!-- matrix expression end --> \left( \begin{array}{cc} M & A \\ B & C \end{arry} \right) where M is the Gram matrix of a non-degenerate sesquilinear form and A,B and C are appropriate zero matrices. The dimension of the maximal isotropic subspaces is the sum of the Witt index and the dimension of the radical.
Morphisms of quadratic forms
Consider two formed vector spaces (V,f) and (W,g) over the same field F, where both f and g are quadratic forms. Suppose that \phiφ is a linear map from V to W. The map \phiφ is an isometry Isometry from the formed space (V,f) to the formed space (W,g) if for all v,w in V we have f(v) = f'(\phi(v)). f(v) = f'(φ(v)). The map \phiφ is a similarity Similarity from the formed space (V,f) to a formed space (W,g) if for all v,w in V we have f(v) = \lambda f'(\phi(v)). f(v) = λ (f'(φ(v))) for some non-zero λ in F\lambda \in F. Finally, the map φ\phi. is a semi-similarity Semi-similarity from the formed space (V,f) to the formed space (W,g) if for all v,w in V we have f(v)=\lambda f'(\phi(v))^\alpha f(v) = λ f'(φ(v))<SUP>α</SUP> for some non-zero λ in F\lambda \in F and a field automorphism α\alpha of F.

Also in this case, one of the objectives of studying maps between formed vector spaces is the classification of quadratic forms of the same vector space V, where it is sufficient to classify non-degenerate forms.

Since there is a one-to-one relationship between quadratic forms in odd characteristic and orthogonal bilinear forms in odd characteristic, we suppose in this section that f is a quadratic form in even characteristic. We call a two dimensional vector space a hyperbolic line if it contains a non-zero vector such that f(v) = 0. It is proved (see Proposition 6.9 of ) that any two hyperbolic lines are isometric, and we choose as canonical representative the quadratic form with polynomial x_1 x_2x<sub>1</sub>x<sub>2</sub>. As in the case of the orthogonal bilinear forms, it can be proved (see Theorem 6.10 of ) that (V,f) can be written as the orthogonal direct sum of hyperbolic lines and one subspace U of dimension at most two. The behaviour of f on the subspace U determines the similarity class of f. The classification of quadratic forms in even characteristic is analogous to the one in odd characteristic. If the dimension of U is zero, then (V,f) is the orthogonal direct sum of hyperbolic lines, and hence (V,f) is isometric to the formed space (V,f'), with polynomial x_1 x_2 + \ldots + x_{n-1}x_n, x<sub>1</sub> x<sub>2</sub> + ... + x<sub>n<font face="symbol">-</font >1</sub>x<sub>n</sub>, which is chosen as the canonical form. Note that the dimension of the vector space V is necessarily even. We call f hyperbolic (see also Section ). It follows also that in this similarity class, there is only one isometry class. The Witt index of f equals n/2. If the dimension of U is one, then necessarily the polynomial of f equals \mu x_1^2 + x_2 x_3 + \ldots + x_{n-1}x_n μ x<sub>1</sub><sup>2</sup> + x<sub>2</sub> x<sub>3</sub> + ... + x<sub>n<font face="symbol">-</font >1</sub>x<sub>n</sub> for some \mu \in Fμ in F, and the dimension of the vector space V is odd. We call f parabolic (see also Section ). Since every element is a square in even characteristic, rescaling the first basis vector yields \mu=1μ=1. The Witt index of f equals (n-1)/2. Suppose at last that the dimension of U is two. We may suppose that U is not a hyperbolic line. It is not difficult to see that a suitable base change yields the polynomial d x_1^2 + x_1x_2 + x_2^2 + x_3 x_4 + \ldots + x_{n-1}x_n d x<sub>1</sub><sup>2</sup> + x<sub>1</sub>x<sub>2</sub> + x<sub>2</sub><sup>2</sup> + x<sub>3</sub> x<sub>4</sub> + ... + x<sub>n<font face="symbol">-</font>1</sub>x<sub>n</sub> for an element of category 1, this is, an element d such that T(d)=1 with T the trace map from F to GF(2). Furthermore, an easy argument shows that an appropriate base change allows to choose any element of category 1 for d. It follows also that the dimension of the vector space V is even. We call f elliptic (see also Section ). The Witt index of f equals (n-2)/2. Hence, non-degenerate quadratic forms in even characteristic come in three similarity classes, which is analogous to the odd characteristic case, and each similarity class contains only one isometry class, which is different than in the odd characteristic case

Suppose that f is a degenerate quadratic form on the n-dimensional vector space V, then Rad(f) is a non-trivial subspace of the vector space V. The vector space V can be written as the orthogonal direct sum of a subspace W and Rad(f), and the restriction of f to W is a non-degenerate quadratic form on W. Hence, f is isometric with a quadratic form with one of the three above polynomials. The dimension of the maximal isotropic subspaces is the sum of the Witt index and the dimension of the radical.

Operations based on morphisms of forms
a transition matrix b from one basis to another
The argument f is a sesquilinear or quadratic form. For every isometry class of forms, there is a canonical representative, as described in Section . If M is the Gram matrix of the form f, then this method returns an invertible matrix b such that b * M * TransposedMat(b) is the Gram matrix of the canonical representative. That is, b is the transition matrix from a basis of the underlying vector space of f to another basis. <#Include SYSTEM "../examples/include/basechangetocanonical.include">
the inner automorphism of GL(d,q) associated to the transition matrix b.
The argument b must be an invertible matrix of size d over the finite field gf of order q. This method returns the inner automorphism of GL(d,q) induces by conjugation by b. <#Include SYSTEM "../examples/include/basechangehom.include">
the canonical form isometric to the sesquilinear or quadratic form f.
The argument f is a sesquilinear or quadratic form. For every isometry class of forms, there is a canonical representative, as described in Section , which is the returned form. <#Include SYSTEM "../examples/include/isometriccanonicalform.include">
a finite field element
Recall that a similarity of a form f on a vector space V, is a linear transformation g of V where there exists some nonzero scalar c such that for all v,w in V, f(u^g,v^g) = c f(u,v). f(u<sup>g</sup>,v<sup>g</sup>) = c f(u,v) This operation finds for a particular matrix M, giving rise to a similarity of the sesquilinear form form, the said scalar c. <#Include SYSTEM "../examples/include/scalarfromsim.include">
the Witt index of the form f.
The argument f is a sesquilinear or quadratic form. When f is degenerate, its Witt index is defined as the Witt index of its non-degenerate part, see Sections and .
true or false.
The argument f is a sesquilinear or quadratic form. This operation returns true is and only if f is elliptic; that is, it is orthogonal of minus type, or in other words, has even dimension and non-maximal Witt index (see Section for sesquilinear forms and Section for quadratic forms).
true or false.
The argument f is a sesquilinear or quadratic form. This operation returns true is and only if f is parabolic; that is, it is orthogonal of neutral type, or in other words, it has odd dimension (see Section for sesquilinear forms and Section for quadratic forms).
true or false.
The argument f is a sesquilinear or quadratic form. This operation returns true is and only if f is hyperbolic; that is, it is orthogonal of plus type, or in other words, has even dimension and maximal Witt index (see Section for sesquilinear forms and Section for quadratic forms).