<!-- 

  theory.xml    Forms package documentation
                                                                   John Bamberg
                                                               and Jan De Beule
                                                           
  Copyright (C) 2007, Ghent University

This is the chapter of the documentation describing affine spaces.
-->

<Chapter Label="theory">
<Heading>Background Theory on Forms</Heading>

In this section, we give a very brief overview on the theory
of sesquilinear and quadratic forms. The reader can find more in the
texts: Cameron <Cite Key="Cameron"/>, Taylor <Cite Key="Taylor"/>,
Aschbacher <Cite Key="Aschbacher"/>, or Kleidman and Liebeck <Cite Key="KleidmanLiebeck"/>.

<Section>
<Heading>Sesquilinear forms, dualities, and polarities</Heading>
A <E>sesquilinear form</E><Index Key="Form" Subkey="Sesquilinear">Form</Index> on a vector space <M>V</M> over a field <M>F</M>, is a map
<M>f</M> from <M>V\times V</M> to <M>F</M> which is linear in the first coordinate, but
semilinear<Index>Semilinear</Index> in the second coordinate; that is, there is a field
automorphism <M>\alpha</M> (the <E>companion automorphism</E><Index>Companion Automorphism</Index> of <M>f</M>) such
that <M>f(v,\lambda w)=\lambda^\alpha f(v,w)</M> for all <M>v,w\in V</M> and <M>\lambda\in F</M>.
If <M>\alpha</M> is the identity, then <M>f</M> is <E>bilinear</E><Index Key="Form" Subkey="Bilinear">Form</Index>. Two vectors <M>v</M> and
<M>w</M> are <E>orthogonal</E><Index>Orthogonal</Index> (w.r.t. <M>f</M>) if <M>f(v,w) = 0</M>. 
The <E>radical</E><Index>Radical</Index> of <M>f</M>
is the subspace consisting of vectors which are orthogonal to every vector, and we say that <M>f</M> is
<E>non-degenerate</E><Index Key="Form" Subkey="Non-degenerate">Form</Index> if its radical is trivial (and <E>degenerate</E> otherwise).
A <E>duality</E><Index>Duality</Index> <M>\delta</M> of a projective space <M>\mathcal{P}</M> is an incidence reversing permutation
of the subspaces of <M>\mathcal{P}</M>, and a <E>polarity</E><Index>Polarity</Index> of <M>\mathcal{P}</M> is a duality of order 2.
An example of such arises from a non-degenerate sesquilinear form; given
a subspace <M>W</M>, we let <M>W^\perp</M> be the set of points which are orthogonal 
with every element of <M>W</M>. We say that a subspace <M>W</M> is <E>totally isotropic</E><Index>Totally Isotropic</Index>
with respect to a polarity if <M>W</M> contains or is contained in <M>W^\perp</M>.

The Birkhoff-von Neumann Theorem<Index>Birkhoff-von Neumann Theorem</Index> 
states that every duality of the projective space <M>PG(n,q)</M> arises from a non-degenerate sesquilinear form
(up to a scalar). Such a duality is a polarity if it is <E>reflexive</E><Index Key="Form" Subkey="Reflexive">Form</Index>, i.e., 
<M>f(v,w)=0</M> implies <M>f(w,v)=0</M>.

Now a sesquilinear form <M>f</M> is <E>hermitian</E><Index Key="Form" Subkey="Hermitian">Form</Index> if <M>f(v,w)=f(w,v)^\alpha</M> holds where
<M>\alpha</M> is the companion automorphism of <M>f</M> and <M>\alpha</M> has order 2. But if
<M>\alpha</M> is trivial then <M>f</M> is <E>symmetric</E><Index Key="Form" Subkey="Symmetric">Form</Index>. If <M>f</M> inturn satisfies <M>f(v,v)=0</M> 
(for all <M>v</M>) then <M>f</M> is <E>alternating</E><Index Key="Form" Subkey="Alternating">Form</Index>. It is a well-known theorem of polar
geometry that a non-degenerate reflexive sesquilinear form is either
alternating, symmetric, or similar to an hermitian form.
The associated polarity is called <E>symplectic</E><Index Key="Polarity" Subkey="Symplectic">Polarity</Index>, 
<E>orthogonal</E><Index Key="Polarity" Subkey="Orthogonal">Polarity</Index>, and <E>unitary</E><Index Key="Polarity" Subkey="Unitary">Polarity</Index>
respectively (though there are some other conventions for the characteristic 2 case). 

<Subsection>
<Heading>Example</Heading>
Let <M>M</M> be an invertible 4-dimensional square matrix over <M>F</M> and consider the following
map on pairs of elements of the 4-dimensional vector space <M>V</M> over <M>F</M>:
<Display>f(v, w) = v M w^T.</Display>
Then <M>f</M> is a sesquilinear form of <M>V</M> and <M>M</M> is the <E>Gram matrix</E> of <M>f</M>.
</Subsection>
</Section>

<Section>
<Heading>Quadratic forms</Heading>
We have seen that a polar space can arise from a reflexive sesquilinear form,
but there are other polar spaces which do not arise this way, but instead
have an associated quadratic form. A map <M>Q</M> from a vector space <M>V</M> to
a field <M>F</M> is a <E>quadratic form</E><Index Key="Form" Subkey="Quadratic">Form</Index> if it satisfies
<M>Q(\lambda v) = \lambda^2Q(v)</M>
for all <M>v\in V</M> and <M>\lambda\in F</M>. We say that a subspace <M>W</M> is <E>totally singular</E><Index>Totally Singular</Index>
if the restriction of <M>Q</M> to <M>W</M> is trivial. Note that a subspace is totally isotropic (with respect to 
the associated polarity) if it is totally singular, but the converse is not always true.

<Subsection>
<Heading>Example</Heading>
Let <M>M</M> be an invertible 4-dimensional square matrix over <M>F</M> and consider the following
map on elements of the 4-dimensional vector space <M>V</M> over <M>F</M>:
<Display>f(v) = v M v^T.</Display>
Then <M>f</M> is a quadratic form of <M>V</M> and <M>M</M> is the <E>Gram matrix</E> of <M>f</M>.
</Subsection>

Given a quadratic form <M>Q</M>, there is an associated sesquilinear form
<M>f</M> (which may not be reflexive) defined as follows
<Display>f(v,w)=Q(v+w)-Q(v)-Q(w).</Display>
For characteristic not 2, the quadratic form and its associated sesquilinear form
<M>f</M> determine one another, as <M>2 Q(v)= f(v,v)</M> (for all <M>v</M>).
</Section>

<Section>
<Heading>Morphisms of forms</Heading>
An <E>isometry</E><Index>Isometry</Index> from a formed space <M>(V,f)</M> to a formed space
<M>(W,f')</M> is a bijection <M>\phi</M> such that for all <M>v,w</M> in <M>V</M> we have
<Display>f(v,w) = f'(\phi(v), \phi(w)).</Display>
The weaker notions of <E>similarity</E><Index>Similarity</Index> and <E>semi-similarity</E><Index>Semi-similarity</Index> are also important
in polar geometry. If there exists a scalar <M>\lambda</M> such that for all
<M>v,w</M> in <M>V</M> we have
<Display>f(v,w) = \lambda f'(\phi(v), \phi(w))</Display>
then we say that <M>\phi</M> is a similarity. If we also have a fixed field automorphism
<M>\alpha</M> such that
<Display>f(v,w)=\lambda f'(\phi(v), \phi(w))^\alpha,</Display>
then <M>\phi</M> is a semi-similarity. Naturally, we say that the formed
spaces <M>(V,f)</M> and <M>(W,f')</M> are <E>isometric</E><Index>Isometric</Index> (resp. <E>similar</E><Index>Similar</Index>) if
there exists an isometry (resp. similarity) between them.
Every non-degenerate reflexive sesquilinear form is alternating, symmetric,
or similar to an hermitian form. Thus, up to similarity, the non-degenerate
polar spaces come in five flavours: symplectic, unitary, orthogonal-elliptic<Index Key="Polarity" Subkey="Elliptic">Polarity</Index>,
orthogonal-hyperbolic<Index Key="Polarity" Subkey="Hyperbolic">Polarity</Index>, and 
orthogonal-parabolic<Index Key="Polarity" Subkey="Parabolic">Polarity</Index>. In the case of the
orthogonal spaces, they are distinguished by their Witt Index<Index>Witt Index</Index> (the common
dimension of their maximal totally singular/isotropic subspaces).
</Section>

<Section>
<Heading>An important convention</Heading>
In <Package>Forms</Package>, we have stipulated a convention on
the creation of forms so as to cause as little
confusion as possible. The hermitian forms will simply be those
with the Frobenius Automorphism is the companion automorphism.
We should also caution the user on what information is "enough" to
specify a form as problems can arise in even characteristic.

<Subsection>
<Heading>Example</Heading>
Let <M>F</M> be a finite field of square order and let
<M>M</M> be the following <M>4\times 4</M> matrix over <M>F</M>:
<Table Align="|cccc|">
<HorLine/>
<Row>   <Item>0</Item><Item>1</Item><Item>0</Item><Item>0</Item>  </Row>
<Row>   <Item>1</Item><Item>0</Item><Item>0</Item><Item>0</Item>  </Row>
<Row>   <Item>0</Item><Item>0</Item><Item>0</Item><Item>1</Item>  </Row>
<Row>   <Item>0</Item><Item>0</Item><Item>1</Item><Item>0</Item>  </Row>
<HorLine/>
</Table>
Let <M>\alpha</M> be the unique automorphism of <M>F</M> of order 2. Then
the form
<Display>
(u, v) := uM(v^T)^\alpha
</Display>
defines a non-degenerate hermitian sesquilinear form. If <M>F</M> has odd characteristic,
then the form
<Display>
(u, v) := uMv^T
</Display>
defines a non-degenerate orthogonal form, but if <M>F</M> has even characteristic,
then this form is both:
<Enum>
 <Item>a symplectic bilinear form, and</Item>
 <Item>the associated bilinear form arising from a quadratic form.</Item>
</Enum>
In the latter, case we see that the bilinear form does not <E>define</E> the quadratic form,
but rather that the quadratic form is necessary in order to define the polar geometry.<!-- -->
</Subsection>
</Section>

<Section Label="canonical">
<Heading>Canonical forms</Heading>
Every nondegenerate polar space has a direct decomposition into
a sum
<Display>L_1\perp L_2\perp\cdots L_n\perp U</Display>
where each of the <M>L_i</M> are hyperbolic lines and <M>U</M>
is an anisotropic subspace of dimension at most 2. Thus
if the given polar space is defined by a sesquilinear form
<M>f</M>, then there is an isometric polar space defined by a 
Gram Matrix of the form
<Table Align="|c|cc|cc|cc|cc|cc|cc|">
<HorLine/>
<Row>
<Item>U</Item><Item></Item><Item></Item><Item></Item><Item></Item><Item></Item><Item></Item><Item></Item><Item></Item>
</Row>
<HorLine/>
<Row>
<Item></Item><Item>0</Item><Item>1</Item><Item></Item><Item></Item><Item></Item><Item></Item><Item></Item><Item></Item>
</Row>
<Row>
<Item></Item><Item>\epsilon</Item><Item>0</Item><Item></Item><Item></Item><Item></Item><Item></Item><Item></Item><Item></Item>
</Row>
<HorLine/>
<Row>
<Item></Item><Item></Item><Item></Item><Item>0</Item><Item>1</Item><Item></Item><Item></Item><Item></Item><Item></Item>
</Row>
<Row>
<Item></Item><Item></Item><Item></Item><Item>\epsilon</Item><Item>0</Item><Item></Item><Item></Item><Item></Item><Item></Item>
</Row>
<HorLine/>
<Row>
<Item></Item><Item></Item><Item></Item><Item></Item><Item></Item><Item>*</Item><Item></Item><Item></Item><Item></Item>
</Row>
<Row>
<Item></Item><Item></Item><Item></Item><Item></Item><Item></Item><Item></Item><Item>*</Item><Item></Item><Item></Item>
</Row>
<HorLine/>
<Row>
<Item></Item><Item></Item><Item></Item><Item></Item><Item></Item><Item></Item><Item></Item><Item>0</Item><Item>1</Item>
</Row>
<Row>
<Item></Item><Item></Item><Item></Item><Item></Item><Item></Item><Item></Item><Item></Item><Item>\epsilon</Item><Item>0</Item>
</Row>
<HorLine/>
</Table>
were the top left hand corner represents the anisotropic part, and there are
zeros everywhere else. The value of <M>\epsilon</M> is -1 if the form is alternating,
otherwise it is 1.
</Section>

</Chapter>
