<!-- 
polaritiesofps.xml    FinInG package documentation
                                                                   John Bamberg
                                                                   Anton Betten
                                                                  Philippe Cara
                                                                   Jan De Beule
                                                                 Michel Lavrauw
                                                                Max Neunhoeffer

  Copyright (C) 2018,	Colorado State University
                        Sabancı Üniversitesi
						Università degli Studi di Padova
						Universiteit Gent
						University of St. Andrews
						University of Western Australia
                        Vrije Universiteit Brussel

This is the chapter of the documentation describing the polarities of projective
spaces.

-->

<Chapter Label="polaritiesofps">
<Heading>Polarities of Projective Spaces</Heading>

A <E>polarity</E> of a incidence structure is
an incidence reversing, bijective, and involutory map on the elements of the
incidence structure. It is well known that every polarity of a projective space
is just an involutory correlation of the projective space. The construction of
correlations of a projective space is described in Chapter <Ref Chap="projgroup"/>.

In this chapter we describe methods and operations dealing with the construction
and use of polarities of projective spaces in <Package>FinInG</Package>. 

<Section>
<Heading>Creating polarities of projective spaces</Heading>

Since polarities of a projective space necessarily have an involutory field
automorphism as companion automorphism and the standard duality of the projective
space as the companion projective space isomorphism, a polarity of a projective
space is determined completely by a suitable matrix <M>A</M>. Every polarity of
a projective space &pgnq; is listed in the following table, including
the conditions on the matrix <M>A</M>.  

<Table Align="|l|l|l|">
 <Caption>polarities of a projective space</Caption>
 <HorLine/>
  <Row><Item> </Item><Item><M>q</M> odd</Item><Item><M>q</M> even</Item></Row>
 <HorLine/>
   <Row><Item>hermitian</Item><Item> 
   <!--<Alt Not="HTML"><M>A^{\theta}=A^T</M></Alt><Alt Only="HTML">A&lt;sup>&#952; &lt;/sup> = A&lt;sup>T &lt;/sup></Alt>-->
   <Alt Not="HTML"><M>A^{\theta}=A^T</M></Alt>
   <Alt Only="HTML MathJax"><M>A^{\theta}=A^T</M></Alt>
   <Alt Only="HTML noMathJax">A&lt;sup>&#952; &lt;/sup> = A&lt;sup>T &lt;/sup></Alt>
   </Item>
   <Item>
   <Alt Not="HTML"><M>A^{\theta}=A^T</M></Alt>
   <Alt Only="HTML MathJax"><M>A^{\theta}=A^T</M></Alt>
   <Alt Only="HTML noMathJax">A&lt;sup>&#952; &lt;/sup> = A&lt;sup>T &lt;/sup></Alt>
   </Item></Row>
   <Row><Item>symplectic</Item><Item>
   <Alt Not="HTML"><M>A^T=-A</M></Alt>
   <Alt Only="HTML MathJax"><M>A^T=-A</M></Alt>
   <Alt Only="HTML noMathJax">A&lt;sup>T &lt;/sup> = - A</Alt>
   </Item>
   <Item>
   <Alt Not="HTML"><M>A^T=A</M>, all <M>a_{ii}=0</M></Alt>
   <Alt Only="HTML MathJax"><M>A^T=A</M>, all <M>a_{ii}=0</M></Alt>
   <Alt Only="HTML noMathJax">A&lt;sup>T &lt;/sup> = A, all a&lt;sub>ii&lt;/sub>=0</Alt>
   </Item></Row>
   <Row><Item>orthogonal</Item><Item>
   <Alt Not="HTML"><M>A^T=A</M></Alt>
   <Alt Only="HTML MathJax"><M>A^T=A</M></Alt>
   <Alt Only="HTML noMathJax">A&lt;sup>T &lt;/sup> = A</Alt>
   </Item><Item></Item></Row>
   <Row><Item>pseudo</Item><Item> </Item>
   <Item>
   <Alt Not="HTML"><M>A^T=A</M>, not all <M>a_{ii}=0</M></Alt>
   <Alt Only="HTML MathJax"><M>A^T=A</M>, not all <M>a_{ii}=0</M></Alt>
   <Alt Only="HTML noMathJax">A&lt;sup>T &lt;/sup> = A, not all a&lt;sub>ii&lt;/sub>=0</Alt>
   </Item></Row>
 <HorLine/>
</Table>
A hermitian polarity of the projective space &pgnq; exists if and only
if the field &gfq; admits an involutory field automorphism.<P/>
It is well known that there is a correspondence between polarities of projective
spaces and non-degenerate sesquilinear forms on the underlying vector space.
Consider a sesquilinear form <M>f</M> on the vector space <M>V(n+1,q)</M>. Then
<M>f</M> induces a map on the elements of &pgnq; as follows: every
element with underlying subspace <M>\alpha</M> is mapped to the
element with underlying subspace <Alt Not="HTML"><M>\alpha^{\perp}</M></Alt>
<Alt Only="HTML MathJax"><M>\alpha^\perp</M></Alt><Alt Only="HTML noMathJax">&#945;&lt;sup>&#8869;&lt;/sup></Alt>,
i.e. the subspace of <M>V(n+1,q)</M> orthogonal to <M>\alpha</M> with
respect to the form <M>f</M>. It is clear that
this induced map is a polarity of &pgnq;. Also the converse is true,
with any polarity of &pgnq; corresponds a sesquilinear form on
<M>V(n+1,q)</M>. The above classification of polarities of &pgnq; follows
from the classification of sesquilinear forms on <M>V(n+1,q)</M>. For more
information, we refer to <Cite Key="HirschfeldThas"/> and <Cite
Key="KleidmanLiebeck"/>. We mention that the implementation of the action of
correlations on projective points (see <Ref Sect="projgroup_actions"/>)
guarantees that a sesquilinear form with matrix <M>M</M> and field
automorphism <M>\theta</M> corresponds to a polarity with matrix <M>M</M>
and field automorphism <M>\theta</M> and vice versa.<P/>

In <Package>FinInG</Package>, polarities of projective spaces are always
objects in the category <C>IsPolarityOfProjectiveSpace</C>, which is a
subcategory of the category <C>IsProjGrpElWithFrobWithPSIsom</C>. 

<ManSection>
<Oper Name="PolarityOfProjectiveSpace" Arg="mat, f"/>
<Returns>a polarity of a projective space</Returns>
<Description>The underlying correlation of
the projective space is constructed using matrix <A>mat</A>, field <A>f</A>, the identity
mapping as field automorphism and the standard duality of the projective space.
It is checked whether the matrix <A>mat</A> satisfies the necessary conditions to
induce a polarity.
<Example>
<#Include SYSTEM "../examples/include/polarities_construct1.include"> 
</Example>
</Description>
</ManSection>

<ManSection>
<Oper Name="PolarityOfProjectiveSpace" Arg="mat, frob, f"/>
<Oper Name="HermitianPolarityOfProjectiveSpace" Arg="mat, f"/>
<Returns>a polarity of a projective space</Returns>
<Description>The underlying correlation of
the projective space is constructed using matrix <A>mat</A>, field
automorphism <A>frob</A>, <A>f</A> and the standard duality of the projective space. It is
checked whether the <A>mat</A> satisfies the necessary conditions to 
induce a polarity, and whether <A>frob</A> is a non-trivial involutory field
automorphism. The second operation only needs the arguments <A>mat</A> and
<A>f</A> to construct a hermitian polarity of a projective space, provided the
field <A>f</A> allows an involutory field automorphism and <A>mat</A> satisfies
the necessary conditions. The latter is checked by constructing the underlying
hermitian form.
<Example>
<#Include SYSTEM "../examples/include/polarities_construct2.include"> 
</Example>
</Description>
</ManSection>

<ManSection>
<Oper Name="PolarityOfProjectiveSpace" Arg="form"/>
<Returns>a polarity of a projective space</Returns>
<Description>The polarity of the projective space is constructed using a
non-degenerate sesquilinear form <A>form</A>. It is checked whether the given
form is non-degenerate.
<Example>
<#Include SYSTEM "../examples/include/polarities_fromform.include"> 
</Example>
</Description>
</ManSection>

<ManSection>
<Oper Name="PolarityOfProjectiveSpace" Arg="ps"/>
<Returns>a polarity of a projective space</Returns>
<Description>The polarity of the projective space is constructed using the
non-degenerate sesquilinear form that defines
the polar space <A>ps</A>. When <A>ps</A> is a parabolic
quadric in even characteristic, no polarity of the ambient
projective space can be associated to <A>ps</A>, and an
error message is returned.
<Example>
<#Include SYSTEM "../examples/include/polarities_frompolarspace.include"> 
</Example>
</Description>
</ManSection>


</Section>

<Section>
<Heading>Operations, attributes and properties for polarities of projective spaces</Heading>

<ManSection>
<Attr Name="SesquilinearForm" Arg="phi"/>
<Returns>a sesquilinear form</Returns>
<Description>The sesquilinear form corresponding to the given polarity <A>phi</A> is returned.
<Example>
<#Include SYSTEM "../examples/include/polarities_toform.include"> 
</Example>
</Description>
</ManSection>

<ManSection>
<Attr Name="BaseField" Arg="phi"/>
<Returns>a field</Returns>
<Description>The base field over which the polarity <A>phi</A> was constructed.
<Example>
<#Include SYSTEM "../examples/include/polarities_basefield.include"> 
</Example>
</Description>
</ManSection>

<ManSection>
<Attr Name="GramMatrix" Arg="phi"/>
<Returns>a matrix</Returns>
<Description>The Gram matrix of the polarity <A>phi</A>.
<Example>
<#Include SYSTEM "../examples/include/polarities_grammatrix.include"> 
</Example>
</Description>
</ManSection>

<ManSection>
<Attr Name="CompanionAutomorphism" Arg="phi"/>
<Returns>a field automorphism</Returns>
<Description>The involutory field automorphism accompanying the polarity <A>phi</A>.
<Example>
<#Include SYSTEM "../examples/include/polarities_automorphism.include"> 
</Example>
</Description>
</ManSection>

<ManSection>
<Prop Name="IsHermitianPolarityOfProjectiveSpace" Arg="phi"/>
<Returns>true or false</Returns>
<Description>The polarity <A>phi</A> is a hermitian polarity of a projective space if
and only if the underlying matrix is hermitian.
<Example>
<#Include SYSTEM "../examples/include/polarities_ishermitian.include"> 
</Example>
</Description>
</ManSection>

<ManSection>
<Prop Name="IsSymplecticPolarityOfProjectiveSpace" Arg="phi"/>
<Returns>true or false</Returns>
<Description>The polarity <A>phi</A> is a symplectic polarity of a projective space if
and only if the underlying matrix is symplectic.
<Example>
<#Include SYSTEM "../examples/include/polarities_issymplectic.include"> 
</Example>
</Description>
</ManSection>

<ManSection>
<Prop Name="IsOrthogonalPolarityOfProjectiveSpace" Arg="phi"/>
<Returns>true or false</Returns>
<Description>The polarity <A>phi</A> is an orthogonal polarity of a projective space if
and only if the underlying matrix is symmetric and the characteristic of the
field is odd.
<Example>
<#Include SYSTEM "../examples/include/polarities_isorthogonal.include"> 
</Example>
</Description>
</ManSection>

<ManSection>
<Prop Name="IsPseudoPolarityOfProjectiveSpace" Arg="phi"/>
<Returns>true or false</Returns>
<Description>The polarity <A>phi</A> is a pseudo-polarity of a projective space if
and only if the underlying matrix is symmetric, not all elements on the main
diagonal are zero and the characteristic of the field is even.
<Example>
<#Include SYSTEM "../examples/include/polarities_ispseudo.include"> 
</Example>
</Description>
</ManSection>

</Section>

<Section Label="polarties_absolute">
<Heading>Polarities, absolute points, totally isotropic elements and finite classical polar
spaces</Heading>

We already mentioned the equivalence between polarities of &pgnq; and
sesquilinear forms on <M>V(n+1,q)</M>, hence there is a relation between
polarities of &pgnq; and polar spaces induced by sesquilinear forms. The
following concepts express these relations geometrically.<P/>
Suppose that <M>\phi</M> is a
polarity of &pgnq; and that <M>\alpha</M> is an element of &pgnq;. We call
<M>\alpha</M> a <E>totally isotropic element</E> or an <E>absolute element</E>
if and only if <M>\alpha</M> is incident with <Alt Not="HTML"><M>\alpha^\phi</M></Alt>
<Alt Only="HTML MathJax"><M>\alpha^\phi</M></Alt><Alt Only="HTML noMathJax">&#945;&lt;sup>&#966;&lt;/sup></Alt>.
An absolute element that is a
point is also called an <E>absolute point</E> or an <E>isotropic point</E>. It
is clear that an element of &pgnq; is absolute if and only if the
underlying vector space is totally isotropic with respect to the sesquilinear
form equivalent to <M>\phi</M>. Hence the absolute elements induce a
<E>finite classical polar space</E>, the same that is induced by the
equivalent sesquilinear form. When <M>\phi</M> is a pseudo-polarity,
the set of absolute elements are the elements of a hyperplane of &pgnq;.
<P/>
We restrict our introduction to finite classical polar spaces in this
section to the following examples. Many aspects of these geometries
are extensively described in Chapter <Ref Chap="classicalpolarspaces"/>.
<ManSection>
<Oper Name="GeometryOfAbsolutePoints" Arg="f"/>
<Returns>a polar space or a hyperplane</Returns>
<Description>When <A>f</A> is not a pseudo-polarity, this operation returns the
polar space induced by <A>f</A>. When <A>f</A> is a pseudo-polarity, this
operation returns the hyperplane containing all absolute elements.
<Example>
<#Include SYSTEM "../examples/include/polarities_geometryofabsolutepoints.include"> 
</Example>
</Description>
</ManSection>

<ManSection>
<Oper Name="AbsolutePoints" Arg="f"/>
<Returns>a set of points</Returns>
<Description>This operation returns all points that are absolute with respect
to <A>f</A>.
<Example>
<#Include SYSTEM "../examples/include/polarities_absolutepoints.include"> 
</Example>
</Description>
</ManSection>

<ManSection>
<Oper Name="PolarSpace" Arg="f"/>
<Returns>a polar space </Returns>
<Description>When <A>f</A> is not a pseudo-polarity, this operation returns the
polar space induced by <A>f</A>.
<Example>
<#Include SYSTEM "../examples/include/polarities_polarspace.include"> 
</Example>
</Description>
</ManSection>

</Section>

<Section>
<Heading>Commuting polarities</Heading>

<Package>FinInG</Package> constructs polarities of projective
spaces as correlations. This allows polarities to be multiplied
easily, resulting in a collineation. The resulting collineation
is constructed in the correlation group but can be mapped onto its
unique representative in the collineation group. We provide an
example with two commuting polarities.

<Example>
<#Include SYSTEM "../examples/include/polarities_commuting.include"> 
</Example>
</Section>
</Chapter>
