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<a id="X876240A479A5717C" name="X876240A479A5717C"></a>
<div class="ChapSects"><a href="chap10.html#X876240A479A5717C">10 Geometry Morphisms</a>
<div class="ContSect"> <a href="chap10.html#X850559BF7886E0D2">10.1 Geometry morphisms in FinInG</a>

<div class="ContSSBlock">
 <a href="chap10.html#X7A3BDDBE7BADCF0A">10.1-1 IsGeometryMorphism</a>
 <a href="chap10.html#X86E0B2117DD536D2">10.1-2 Intertwiner</a>
</div></div>
<div class="ContSect"> <a href="chap10.html#X7926E5367D0C80B7">10.2 Type preserving bijective geometry morphisms</a>

<div class="ContSSBlock">
 <a href="chap10.html#X8774AF5B840E488D">10.2-1 IsomorphismPolarSpaces</a>
</div></div>
<div class="ContSect"> <a href="chap10.html#X79C677CD7B7EC451">10.3 Klein correspondence and derived dualities</a>

<div class="ContSSBlock">
 <a href="chap10.html#X7C8E719883F407BB">10.3-1 PluckerCoordinates</a>
 <a href="chap10.html#X7CD8E5187C5B7116">10.3-2 KleinCorrespondence</a>
 <a href="chap10.html#X7CD8E5187C5B7116">10.3-3 KleinCorrespondence</a>
 <a href="chap10.html#X7FE537C57F2F6BAA">10.3-4 KleinCorrespondenceExtended</a>
 <a href="chap10.html#X8472EBA584544B51">10.3-5 NaturalDuality</a>
 <a href="chap10.html#X7954DA578247B06F">10.3-6 SelfDuality</a>
</div></div>
<div class="ContSect"> <a href="chap10.html#X86D21DCB7C0029F9">10.4 Embeddings of projective spaces</a>

<div class="ContSSBlock">
 <a href="chap10.html#X857976B679B8EC30">10.4-1 NaturalEmbeddingBySubspace</a>
 <a href="chap10.html#X82510EF078F0806E">10.4-2 NaturalEmbeddingBySubField</a>
 <a href="chap10.html#X7BC7FCDC7D9E1A09">10.4-3 Embedding of projective spaces by field reduction</a>

 <a href="chap10.html#X7F84986879DD952A">10.4-4 BlownUpSubspaceOfProjectiveSpace</a>
 <a href="chap10.html#X7D2D9AD287532FDB">10.4-5 NaturalEmbeddingByFieldReduction</a>
</div></div>
<div class="ContSect"> <a href="chap10.html#X7C00DD48787B1EEE">10.5 Embeddings of polar spaces</a>

<div class="ContSSBlock">
 <a href="chap10.html#X857976B679B8EC30">10.5-1 NaturalEmbeddingBySubspace</a>
 <a href="chap10.html#X82510EF078F0806E">10.5-2 NaturalEmbeddingBySubField</a>
 <a href="chap10.html#X7823BA95797898CE">10.5-3 Embedding of polar spaces by field reduction</a>

 <a href="chap10.html#X7D2D9AD287532FDB">10.5-4 NaturalEmbeddingByFieldReduction</a>
 <a href="chap10.html#X7D2D9AD287532FDB">10.5-5 NaturalEmbeddingByFieldReduction</a>
</div></div>
<div class="ContSect"> <a href="chap10.html#X81FAC1DE7C4B1972">10.6 Projections</a>

<div class="ContSSBlock">
 <a href="chap10.html#X7B9E8AB682EC07C2">10.6-1 NaturalProjectionBySubspace</a>
</div></div>
<div class="ContSect"> <a href="chap10.html#X7952EE1A80D53825">10.7 Projective completion</a>

<div class="ContSSBlock">
 <a href="chap10.html#X7E23CA137AEFD1D2">10.7-1 ProjectiveCompletion</a>
</div></div>
</div>

<h3>10 Geometry Morphisms</h3>

Here we describe what is meant by a geometry morphism in FinInG and the various operations and tools available to the user. When using groups in GAP, we often use homomorphisms to pass from one situation to another, even though mathematically it may appear to be unnecessary, there can be ambiguities if the functionality is too flexible. This also applies to finite geometry. Take for example the usual exercise of thinking of a hyperplane in a projective space as another projective space. To conform with similar situations in GAP, the right thing to do is to embed one projective space into another, rather than having one projective space automatically as a substructure of another. The reason for this is that there are many ways one can do this embedding, even though we may dispense with this choice when we are working mathematically. So to avoid ambiguity, we stipulate that one should construct the embedding explicitly. How this is done will be described this chapter.

Suppose that S and S' are two incidence geometries. A geometry morphism from S to S' is defined to be a map from the elements of S to the elements of S' which preserves incidence and induces a function from the type set of S to the type set of S'. For instance, a correlation and a collineation are examples of geometry morphisms, but they have been dealt with in more specific ways in FinInG. We will mainly be concerned with geometry morphisms where the source and range are different. Hence, the natural embedding of a projective space in a larger projective space, the mapping induced by field reduction, and e.g. the Klein correspondence are examples of such geometry morphisms.

As a geometry morphism from S to S' preserves incidence, it also preserves the symmetry, and hence it induces also a map from the collineation group of S into the collineation group of S'. Such a map will be called an Intertwiner, and FinInG can provide these maps for some of the geometry morphisms.

Note that quite some technicalities are needed in the implementation of some geometry morphisms. This chapters deals only with the user interface. Some low level functions for geometry morphisms are described in Appendix <a href="chapC.html#X874D94F47C943D71">C</a>.

<a id="X850559BF7886E0D2" name="X850559BF7886E0D2"></a>

<h4>10.1 Geometry morphisms in FinInG</h4>

<a id="X7A3BDDBE7BADCF0A" name="X7A3BDDBE7BADCF0A"></a>

<h5>10.1-1 IsGeometryMorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGeometryMorphism</code></td><td class="tdright">( family )</td></tr></table></div>
The category <code class="code">IsGeometryMorphism</code> represents a special object in FinInG which carries attributes and the given element map. The element map is given as a <code class="code">IsGeneralMapping</code>, and so has a source and range.

<div class="example"><pre>
gap> ShowImpliedFilters(IsGeometryMorphism);
Implies:
 IsGeneralMapping
 IsTotal
 Tester(IsTotal)
 IsSingleValued
 Tester(IsSingleValued)

</pre></div>

The usual operations of <code class="code">ImageElm</code>, and <code class="code">PreImageElm</code>, have methods installed for geometry morphisms, as well as the overload operator <code class="code">\^</code>.

<a id="X86E0B2117DD536D2" name="X86E0B2117DD536D2"></a>

<h5>10.1-2 Intertwiner</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Intertwiner</code>( <var class="Arg">f</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: a group homomorphism

The argument <var class="Arg">f</var> is a geometry morphism. If <var class="Arg">f</var> comes equipped with a natural intertwiner from an automorphism group of the source of <var class="Arg">f</var> to the automorphism group to the image of <var class="Arg">f</var>, then the user may is able to obtain the intertwiner by calling this operation (see the individual geometry morphism constructions). For most geometry morphisms, there is also an accompanying intertwiner for the automorphism groups of the source and range. Given a geometry morphism f from S to S', an intertwiner φ is a map from the automorphism group of S to the automorphism group of S', such that for every element p of S and every automorphism g of S, we have

f(pg)=f(p)φ(g).

There is no method to compute an intertwiner for a given geometry morphism, the attribute is or is not set during the construction of the geometry morphism, depending whether the Source and Range of the morphism have the appropriate automorphism group known as an attribute. When this condition is not satisfied, the user is expected to call the appropriate automorphism groups, so that they are computed, and to recompute the geometry morphism (which will not cost a lot of computation time then). This will make the attribute <code class="file">Intertwiner</code> available. Here is a simple example of the intertwiner for the isomorphism of two polar spaces (see <code class="func">IsomorphismPolarSpaces</code> (<a href="chap10.html#X8774AF5B840E488D">10.2-1</a>)). The source of the homomorphism is dependent on the geometry.

<div class="example"><pre>
gap> form := BilinearFormByMatrix( IdentityMat(3,GF(3)), GF(3) );
< bilinear form >
gap> ps := PolarSpace(form);
<polar space in ProjectiveSpace(2,GF(3)): x_1^2+x_2^2+x_3^2=0 >
gap> pq := ParabolicQuadric(2,3);
standard Q(2, 3)
gap> iso := IsomorphismPolarSpaces(ps, pq);
#I Computing nice monomorphism...
<geometry morphism from <Elements of <polar space in ProjectiveSpace(2,GF(
3)): x_1^2+x_2^2+x_3^2=0 >> to <Elements of standard Q(2, 3)>>
gap> KnownAttributesOfObject(iso);
[ "Range", "Source", "Intertwiner" ]
gap> hom := Intertwiner(iso);
MappingByFunction( <projective semilinear group with
3 generators>, PGammaO(3,3), function( y ) ... end, function( x ) ... end )

</pre></div>

<a id="X7926E5367D0C80B7" name="X7926E5367D0C80B7"></a>

<h4>10.2 Type preserving bijective geometry morphisms</h4>

An important class of geometry morphisms in FinInG are the isomorphisms between polar spaces of the same kind that are induced by coordinate transformations.

<a id="X8774AF5B840E488D" name="X8774AF5B840E488D"></a>

<h5>10.2-1 IsomorphismPolarSpaces</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismPolarSpaces</code>( <var class="Arg">ps1</var>, <var class="Arg">ps2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismPolarSpaces</code>( <var class="Arg">ps1</var>, <var class="Arg">ps2</var>, <var class="Arg">boolean</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a geometry morphism

The arguments <var class="Arg">ps1</var> and <var class="Arg">ps2</var> are equivalent polar spaces, i.e. up to coordinate transformation, the underlying sesquilinear or quadratic form determines the same polar space, or, <var class="Arg">ps1</var> is a parabolic quadric over a finite field f of even characteristic in dimension 2n and <var class="Arg">ps2</var> is a symplectic space over f in dimension 2n-1, then this operation returns a geometry isomorphism between them. The optional third argument <var class="Arg">boolean</var> can take either <code class="code">true</code> or <code class="code">false</code> as input, and then the operation will or will not compute the intertwiner accordingly. The user may wish that the intertwiner is not computed when working with large polar spaces. The default (when calling the operation with two arguments) is set to <code class="code">true</code>, and in this case, if at least one of <var class="Arg">ps1</var> or <var class="Arg">ps2</var> has a collineation group installed as an attribute, then an intertwining homomorphism is installed as an attribute of the resulting geometry morphism. Hence we also obtain a natural group isomorphism from the collineation group of <var class="Arg">ps1</var> onto the collineation group of <var class="Arg">ps2</var> (see also <code class="func">Intertwiner</code> (<a href="chap10.html#X86E0B2117DD536D2">10.1-2</a>)).

<div class="example"><pre>
gap> mat1 := IdentityMat(6,GF(5));
< mutable compressed matrix 6x6 over GF(5) >
gap> form1 := BilinearFormByMatrix(mat1,GF(5));
< bilinear form >
gap> ps1 := PolarSpace(form1);
<polar space in ProjectiveSpace(
5,GF(5)): x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2=0 >
gap> mat2 := [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],
> [0,0,0,0,0,0],[0,0,0,0,0,0],[0,0,0,0,0,0]]*Z(5)^0;
[ [ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), Z(5)^0 ],
 [ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), Z(5)^0, 0*Z(5) ],
 [ 0*Z(5), 0*Z(5), 0*Z(5), Z(5)^0, 0*Z(5), 0*Z(5) ],
 [ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5) ],
 [ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5) ],
 [ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5) ] ]
gap> form2 := QuadraticFormByMatrix(mat2,GF(5));
< quadratic form >
gap> ps2 := PolarSpace(form2);
<polar space in ProjectiveSpace(5,GF(5)): x_1*x_6+x_2*x_5+x_3*x_4=0 >
gap> iso := IsomorphismPolarSpaces(ps1,ps2,true);
#I No intertwiner computed. One of the polar spaces must have a collineation group computed
<geometry morphism from <Elements of Q+(5,
5): x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2=0> to <Elements of Q+(5,
5): x_1*x_6+x_2*x_5+x_3*x_4=0>>
gap> CollineationGroup(ps1);
#I Computing collineation group of canonical polar space...
<projective collineation group of size 58032000000 with 4 generators>
gap> CollineationGroup(ps2);
#I Computing collineation group of canonical polar space...
<projective collineation group of size 58032000000 with 4 generators>
gap> iso := IsomorphismPolarSpaces(ps1,ps2,true);
<geometry morphism from <Elements of Q+(5,
5): x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2=0> to <Elements of Q+(5,
5): x_1*x_6+x_2*x_5+x_3*x_4=0>>
gap> hom := Intertwiner( iso );
MappingByFunction( <projective collineation group of size 58032000000 with
4 generators>, <projective collineation group of size 58032000000 with
4 generators>, function( y ) ... end, function( x ) ... end )
gap> ps1 := ParabolicQuadric(6,8);
Q(6, 8)
gap> ps2 := SymplecticSpace(5,8);
W(5, 8)
gap> em := IsomorphismPolarSpaces(ps1,ps2);
#I Have 36171 points.
#I Have 37381 points in new orbit.
#I Have 36171 points.
#I Have 37388 points in new orbit.
<geometry morphism from <Elements of Q(6, 8)> to <Elements of W(5, 8)>>
gap> hom := Intertwiner(em);
MappingByFunction( PGammaO(7,8), <projective collineation group of size
27231016821530296320 with
3 generators>, function( el ) ... end, function( el ) ... end )

</pre></div>

<a id="X79C677CD7B7EC451" name="X79C677CD7B7EC451"></a>

<h4>10.3 Klein correspondence and derived dualities</h4>

The Klein correspondence is a well known geometry morphism from the lines of PG(3,q) to the points of a hyperbolic quadric in PG(5,q). This morphism and some derived morphisms are provided in FinInG. The bare essential of the Klein correspondence is the so-called Plücker map.

<a id="X7C8E719883F407BB" name="X7C8E719883F407BB"></a>

<h5>10.3-1 PluckerCoordinates</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PluckerCoordinates</code>( <var class="Arg">line</var> )</td><td class="tdright">( operation )</td></tr></table></div>
This operation takes a line of PG(3,q) as argument. It returns the plucker coordinates of the argument as list of finite field elements. The returned list can be used in operations as <code class="file">vector spaceToElement</code>, and represents a point of the hyperbolic quadric in PG(5,q) with equation X0X5+X1X4+X2X3 = 0

<div class="example"><pre>
gap> pg := PG(3,169);
ProjectiveSpace(3, 169)
gap> l := Random(Lines(pg));
<a line in ProjectiveSpace(3, 169)>
gap> vec := PluckerCoordinates(l);
[ Z(13)^0, Z(13^2)^138, Z(13^2)^93, Z(13^2)^53, Z(13^2)^71, Z(13^2)^106 ]
gap> mat := [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],
> [0,0,0,0,0,0],[0,0,0,0,0,0],[0,0,0,0,0,0]]*Z(13)^0;
[ [ 0*Z(13), 0*Z(13), 0*Z(13), 0*Z(13), 0*Z(13), Z(13)^0 ],
 [ 0*Z(13), 0*Z(13), 0*Z(13), 0*Z(13), Z(13)^0, 0*Z(13) ],
 [ 0*Z(13), 0*Z(13), 0*Z(13), Z(13)^0, 0*Z(13), 0*Z(13) ],
 [ 0*Z(13), 0*Z(13), 0*Z(13), 0*Z(13), 0*Z(13), 0*Z(13) ],
 [ 0*Z(13), 0*Z(13), 0*Z(13), 0*Z(13), 0*Z(13), 0*Z(13) ],
 [ 0*Z(13), 0*Z(13), 0*Z(13), 0*Z(13), 0*Z(13), 0*Z(13) ] ]
gap> form := QuadraticFormByMatrix(mat,GF(169));
< quadratic form >
gap> klein := PolarSpace(form);
<polar space in ProjectiveSpace(5,GF(13^2)): x_1*x_6+x_2*x_5+x_3*x_4=0 >
gap> VectorSpaceToElement(klein,vec);
<a point in Q+(5, 169): x_1*x_6+x_2*x_5+x_3*x_4=0>

</pre></div>

<a id="X7CD8E5187C5B7116" name="X7CD8E5187C5B7116"></a>

<h5>10.3-2 KleinCorrespondence</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ KleinCorrespondence</code>( <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ KleinCorrespondence</code>( <var class="Arg">f</var>, <var class="Arg">boolean</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ KleinCorrespondence</code>( <var class="Arg">q</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ KleinCorrespondence</code>( <var class="Arg">q</var>, <var class="Arg">boolean</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a geometry morphism

The argument <var class="Arg">f</var> is a finite field, the argument <var class="Arg">q</var> is a prime power. The first an the third version use <code class="keyw">true</code> as value for <var class="Arg">boolean</var>. When using <code class="keyw">true</code> as value for the boolean, the intertwiner is computed. This variant of the operation <code class="file">KleinCorrespondence</code> has always as ambient geometry of its range the hyperbolic quadric Q+(5,q) with equation X0X5+X1X4+X2X3 = 0 The returned geometry morphism has the lines of PG(3,q) as source and the points of Q+(5,q) as range.

<div class="example"><pre>
gap> k := KleinCorrespondence( 9 );
<geometry morphism from <lines of ProjectiveSpace(3, 9)> to <points of Q+(5,
9): x_1*x_6+x_2*x_5+x_3*x_4=0>>
gap> Intertwiner(k);
MappingByFunction( The FinInG collineation group PGammaL(4,9), <projective col
lineation group with
3 generators>, function( g ) ... end, function( g ) ... end )
gap> pg := ProjectiveSpace(3, 9);
ProjectiveSpace(3, 9)
gap> AmbientGeometry(Range(k));
Q+(5, 9): x_1*x_6+x_2*x_5+x_3*x_4=0
gap> l := Random( Lines(pg) );
<a line in ProjectiveSpace(3, 9)>
gap> l^k;
<a point in Q+(5, 9): x_1*x_6+x_2*x_5+x_3*x_4=0>

</pre></div>

<a id="X7CD8E5187C5B7116" name="X7CD8E5187C5B7116"></a>

<h5>10.3-3 KleinCorrespondence</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ KleinCorrespondence</code>( <var class="Arg">quadric</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ KleinCorrespondence</code>( <var class="Arg">quadric</var>, <var class="Arg">boolean</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a geometry morphism

The argument <var class="Arg">quadric</var> is a hyperbolic quadric in a 5 dimensional projective space. If <var class="Arg">boolean</var> is <code class="keyw">true</code> or not given, this operation returns the geometry morphism equipped with an intertwiner. The returned geometry morphism has the lines of PG(3,q) as source and the points of Q+(5,q) as range.

<div class="example"><pre>
gap> quadric := HyperbolicQuadric(5,3); 
Q+(5, 3)
gap> k := KleinCorrespondence( quadric );
<geometry morphism from <lines of ProjectiveSpace(3, 3)> to <points of Q+(5,
3)>>
gap> pg := ProjectiveSpace(3, 3);
ProjectiveSpace(3, 3)
gap> l := Random( Lines(pg) );
<a line in ProjectiveSpace(3, 3)>
gap> l^k;
<a point in Q+(5, 3)>
gap> id := IdentityMat(6,GF(13));
< mutable compressed matrix 6x6 over GF(13) >
gap> form := QuadraticFormByMatrix(id,GF(13));
< quadratic form >
gap> quadric := PolarSpace(form);
<polar space in ProjectiveSpace(
5,GF(13)): x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2=0 >
gap> k := KleinCorrespondence( quadric );
<geometry morphism from <lines of ProjectiveSpace(3, 13)> to <points of Q+(5,
13): x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2=0>>
gap> pg := AmbientGeometry(Source(k));
ProjectiveSpace(3, 13)
gap> l := Random(Lines(pg));
<a line in ProjectiveSpace(3, 13)>
gap> l^k;
<a point in Q+(5, 13): x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2=0>

</pre></div>

<a id="X7FE537C57F2F6BAA" name="X7FE537C57F2F6BAA"></a>

<h5>10.3-4 KleinCorrespondenceExtended</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ KleinCorrespondenceExtended</code>( <var class="Arg">quadric</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ KleinCorrespondenceExtended</code>( <var class="Arg">quadric</var>, <var class="Arg">boolean</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a geometry morphism

The argument <var class="Arg">quadric</var> is a hyperbolic quadric in a 5 dimensional projective space. If <var class="Arg">boolean</var> is <code class="keyw">true</code> or not given, this operation returns the geometry morphism equipped with an intertwiner. The returned geometry morphism has all the elements of PG(3,q) as source (not just the lines) and the elements of Q+(5,q) as range, hence this operation is a kind of extension of <code class="file">KleinCorrespondence</code>.

<div class="example"><pre>
gap> ps := HyperbolicQuadric(5,7);
Q+(5, 7)
gap> em := KleinCorrespondenceExtended(ps);
<geometry morphism from <All elements of ProjectiveSpace(3,
7)> to <Elements of Q+(5, 7)>>
gap> hom := Intertwiner(em);
MappingByFunction( The FinInG collineation group PGL(4,7), <projective colline
ation group with 2 generators>, function( g ) ... end, function( g ) ... end )
gap> mat := [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],
> [0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]*Z(7)^0;
[ [ 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7), Z(7)^0 ],
 [ 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7), Z(7)^0, 0*Z(7) ],
 [ 0*Z(7), 0*Z(7), 0*Z(7), Z(7)^0, 0*Z(7), 0*Z(7) ],
 [ 0*Z(7), 0*Z(7), Z(7)^0, 0*Z(7), 0*Z(7), 0*Z(7) ],
 [ 0*Z(7), Z(7)^0, 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7) ],
 [ Z(7)^0, 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7) ] ]
gap> g := Projectivity(mat,GF(7));
< a collineation: <cmat 6x6 over GF(7,1)>, F^0>
gap> g in CollineationGroup(ps);
true
gap> PreImageElm(hom,g);
#I <el> is not inducing a collineation of PG(3,q)
fail

</pre></div>

It is well known that the classical generalised quadrangles W(3,q) and Q(4,q) are dual incidence structures, the same holds for the classical generalised quadrangles Q+(5,q) and H(3,q2). Essentially, these dual dualities are based on the Klein correspondence, and are implemented through the operation <code class="file">NaturalDuality</code>, this operation will return a geometry morphism with <code class="code">ElementsOfIncidenceStructure(gq1)</code> as source and <code class="code">ElementsOfIncidenceStructure(gq2)</code> as range, in other words, it is a geometry morphism from all the elements of <var class="Arg">gq1</var> onto all the elements of <var class="Arg">gq2</var>, preserving the incidence, and swapping the types.

<a id="X8472EBA584544B51" name="X8472EBA584544B51"></a>

<h5>10.3-5 NaturalDuality</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalDuality</code>( <var class="Arg">gq1</var>, <var class="Arg">gq2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalDuality</code>( <var class="Arg">gq1</var>, <var class="Arg">gq2</var>, <var class="Arg">boolean</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalDuality</code>( <var class="Arg">gq</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalDuality</code>( <var class="Arg">gq</var>, <var class="Arg">boolean</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a geometry morphism

The operation allows the construction of the duality between W(3,q) and Q(4,q), respectively Q+(5,q), in two directions. It is checked if the arguments are appropriate, i.e. the right type of generalised quadrangle(s). The first version requires two arguments: either the symplectic or parabolic quadrangle, in any order, and defined by any suitable bilinear/quadratic and bilinear form; or the elliptic or hermitian quadrangle (in dimension 3), in any order, and defined by any suitable bilinear/quadratic and hermitian form. In all cases the generalised quadrangles may be the standard one provided by the package FinInG.

The third version requires only one argument, either W(3,q), Q(4,q), Q+(5,q), or H(3,q2), standard or user specified using an appropriate bilinear, quadratic or hermitian form. The range of the returned geometry morphism will be the set of all elements of a suitable generalised quadrangle, in standard form.

The first and third version without a boolean as argument will, if possible return a geometry morphism equipped with an intertwiner. Using the boolean argument <var class="Arg">false</var> will return a geometry morphism that is not equipped with an intertwiner.

<div class="example"><pre>
gap> w := SymplecticSpace(3,5);
W(3, 5)
gap> lines:=AsList(Lines(w));;
gap> duality := NaturalDuality(w);
<geometry morphism from <Elements of W(3, 5)> to <Elements of Q(4, 5)>>
gap> l:=lines[1];
<a line in W(3, 5)>
gap> l^duality;
<a point in Q(4, 5)>
gap> PreImageElm(duality,last);
<a line in W(3, 5)>
gap> hom := Intertwiner(duality);
MappingByFunction( PGammaSp(4,5), <projective collineation group of size
9360000 with 4 generators>, function( g ) ... end, function( g ) ... end )
gap> q := 5;
5
gap> q5q := EllipticQuadric(5,q);
Q-(5, 5)
gap> mat := [[0,1,0,0],[1,0,0,0],[0,0,0,Z(q)],[0,0,Z(q),0]]*Z(q)^0;
[ [ 0*Z(5), Z(5)^0, 0*Z(5), 0*Z(5) ], [ Z(5)^0, 0*Z(5), 0*Z(5), 0*Z(5) ],
 [ 0*Z(5), 0*Z(5), 0*Z(5), Z(5) ], [ 0*Z(5), 0*Z(5), Z(5), 0*Z(5) ] ]
gap> hform := HermitianFormByMatrix(mat,GF(q^2));
< hermitian form >
gap> herm := PolarSpace(hform);
<polar space in ProjectiveSpace(
3,GF(5^2)): x1^5*x2+x1*x2^5+Z(5)*x3^5*x4+Z(5)*x3*x4^5=0 >
gap> duality := NaturalDuality(q5q,herm,true);
<geometry morphism from <Elements of Q-(5, 5)> to <Elements of H(3,
5^2): x1^5*x2+x1*x2^5+Z(5)*x3^5*x4+Z(5)*x3*x4^5=0>>
gap> hom := Intertwiner(duality);
MappingByFunction( PDeltaO-(6,5), <projective collineation group of size
58968000000 with 3 generators>, function( g ) ... end, function( g ) ... end )
gap> g := Random(CollineationGroup(q5q));
< a collineation: <cmat 6x6 over GF(5,1)>, F^0>
gap> g^hom;
< a collineation: <cmat 4x4 over GF(5,2)>, F^5>

</pre></div>

The combination of the isomorphism of the GQs W(3,q), Q(4,q) when q is even and the duality between the same GQs, yields a duality from each of these GQs itself. The operation <code class="file">SelfDuality</code> implements this combination.

<a id="X7954DA578247B06F" name="X7954DA578247B06F"></a>

<h5>10.3-6 SelfDuality</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SelfDuality</code>( <var class="Arg">gq</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SelfDuality</code>( <var class="Arg">gq</var>, <var class="Arg">boolean</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a geometry morphism

It is checked whether the base field of <var class="Arg">gq</var> is a field of characteristic 2 and whether <var class="Arg">gq</var> is a symplectic generalised quadrangle in 3-dimensional projective space or a parabolic quadric in 4-dimensional projective space. The first version will return, when possible, a geometry morphism equipped with an intertwiner. Using the boolean argument <var class="Arg">false</var> will return a geometry morphism that is not equipped with an intertwiner. The example shows the use of the boolean argument.

<div class="example"><pre>
gap> q := 16;
16
gap> mat := [[0,1,0,0,0],[0,0,0,0,0],[0,0,1,0,0],[0,0,0,0,0],[0,0,0,1,0]]*Z(q)^0;
[ [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ] ]
gap> form := QuadraticFormByMatrix(mat,GF(q));
< quadratic form >
gap> q4q := PolarSpace(form);
<polar space in ProjectiveSpace(4,GF(2^4)): x_1*x_2+x_3^2+x_4*x_5=0 >
gap> em := SelfDuality(q4q);
#I No intertwiner computed. The polar space must have a collineation group computed
<geometry morphism from <Elements of Q(4,
16): x_1*x_2+x_3^2+x_4*x_5=0> to <Elements of Q(4,
16): x_1*x_2+x_3^2+x_4*x_5=0>>
gap> CollineationGroup(q4q);
#I Computing collineation group of canonical polar space...
<projective collineation group of size 4380799795200 with 3 generators>
gap> em := SelfDuality(q4q);
<geometry morphism from <Elements of Q(4,
16): x_1*x_2+x_3^2+x_4*x_5=0> to <Elements of Q(4,
16): x_1*x_2+x_3^2+x_4*x_5=0>>
gap> hom := Intertwiner(em);
MappingByFunction( <projective collineation group of size 4380799795200 with
3 generators>, <projective collineation group of size 4380799795200 with
3 generators>, function( el ) ... end, function( el ) ... end )
gap> q := 16;
16
gap> w := SymplecticSpace(3,q);
W(3, 16)
gap> em := SelfDuality(w);
<geometry morphism from <Elements of W(3, 16)> to <Elements of W(3, 16)>>

</pre></div>

<a id="X86D21DCB7C0029F9" name="X86D21DCB7C0029F9"></a>

<h4>10.4 Embeddings of projective spaces</h4>

The most natural of geometry morphisms include, for example, the embedding of a projective space into another via a subspace, the embedding of a projective space over a field into a projective space of the same dimension over an extended field, or the embedding of a projective space over a field into a projective space of higher dimension over a subfield through so-called field reduction.

<a id="X857976B679B8EC30" name="X857976B679B8EC30"></a>

<h5>10.4-1 NaturalEmbeddingBySubspace</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalEmbeddingBySubspace</code>( <var class="Arg">geom1</var>, <var class="Arg">geom2</var>, <var class="Arg">v</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a geometry morphism

The arguments <var class="Arg">geom1</var> and <var class="Arg">geom2</var> are both projective spaces, and <var class="Arg">v</var> is an element of a geom2. This function returns a geometry morphism representing the natural embedding of <var class="Arg">geom1</var> into <var class="Arg">geom2</var> as the subspace <var class="Arg">v</var>. Hence <var class="Arg">geom1</var> and <var class="Arg">v</var> must be equivalent as geometries. An Intertwiner is not implemented for this geometry morphism.

<div class="example"><pre>
gap> geom1 := ProjectiveSpace(2, 3);
ProjectiveSpace(2, 3)
gap> geom2 := ProjectiveSpace(3, 3);
ProjectiveSpace(3, 3)
gap> planes := Planes(geom2);
<planes of ProjectiveSpace(3, 3)>
gap> hyp := Random(planes);
<a plane in ProjectiveSpace(3, 3)>
gap> em := NaturalEmbeddingBySubspace(geom1, geom2, hyp);
<geometry morphism from <All elements of ProjectiveSpace(2,
3)> to <All elements of ProjectiveSpace(3, 3)>>
gap> points := Points(geom1);
<points of ProjectiveSpace(2, 3)>
gap> x := Random(points);
<a point in ProjectiveSpace(2, 3)>
gap> x^em;
<a point in ProjectiveSpace(3, 3)>

</pre></div>

<a id="X82510EF078F0806E" name="X82510EF078F0806E"></a>

<h5>10.4-2 NaturalEmbeddingBySubField</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalEmbeddingBySubField</code>( <var class="Arg">geom1</var>, <var class="Arg">geom2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalEmbeddingBySubField</code>( <var class="Arg">geom1</var>, <var class="Arg">geom2</var>, <var class="Arg">boolean</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a geometry morphism

The arguments <var class="Arg">geom1</var> and <var class="Arg">geom2</var> are projective spaces of the same dimension. This function returns a geometry morphism representing the natural embedding of <var class="Arg">geom1</var> into <var class="Arg">geom2</var> as a subfield geometry. The geometry morphism also comes equipped with an intertwiner (see <code class="func">Intertwiner</code> (<a href="chap10.html#X86E0B2117DD536D2">10.1-2</a>)). The optional third argument <var class="Arg">boolean</var> can take either <code class="code">true</code> or <code class="code">false</code> as input, and then our operation will or will not compute the intertwiner accordingly. The default (when calling the operation with two arguments) is set to <code class="code">true</code>. Note that the source of the intertwiner is the projectivity group of <var class="Arg">geom1</var> and its range is a subgroup of the projectivity group of <var class="Arg">geom2</var>. Here is a simple example where the geometry morphism embeds PG(2,3) into PG(2,9) .

<div class="example"><pre>
gap> pg1 := PG(2,3);
ProjectiveSpace(2, 3)
gap> pg2 := PG(2,9);
ProjectiveSpace(2, 9)
gap> em := NaturalEmbeddingBySubfield(pg1,pg2);
<geometry morphism from <All elements of ProjectiveSpace(2,
3)> to <All elements of ProjectiveSpace(2, 9)>>
gap> points := AsList(Points( pg1 ));
[ <a point in ProjectiveSpace(2, 3)>, <a point in ProjectiveSpace(2, 3)>,
 <a point in ProjectiveSpace(2, 3)>, <a point in ProjectiveSpace(2, 3)>,
 <a point in ProjectiveSpace(2, 3)>, <a point in ProjectiveSpace(2, 3)>,
 <a point in ProjectiveSpace(2, 3)>, <a point in ProjectiveSpace(2, 3)>,
 <a point in ProjectiveSpace(2, 3)>, <a point in ProjectiveSpace(2, 3)>,
 <a point in ProjectiveSpace(2, 3)>, <a point in ProjectiveSpace(2, 3)>,
 <a point in ProjectiveSpace(2, 3)> ]
gap> image := ImagesSet(em, points);
[ <a point in ProjectiveSpace(2, 9)>, <a point in ProjectiveSpace(2, 9)>,
 <a point in ProjectiveSpace(2, 9)>, <a point in ProjectiveSpace(2, 9)>,
 <a point in ProjectiveSpace(2, 9)>, <a point in ProjectiveSpace(2, 9)>,
 <a point in ProjectiveSpace(2, 9)>, <a point in ProjectiveSpace(2, 9)>,
 <a point in ProjectiveSpace(2, 9)>, <a point in ProjectiveSpace(2, 9)>,
 <a point in ProjectiveSpace(2, 9)>, <a point in ProjectiveSpace(2, 9)>,
 <a point in ProjectiveSpace(2, 9)> ]
gap> hom := Intertwiner(em);
MappingByFunction( The FinInG projectivity group PGL(3,3), <projective colline
ation group of size 5616 with
2 generators>, function( x ) ... end, function( y ) ... end )
gap> group1 := ProjectivityGroup(pg1);
The FinInG projectivity group PGL(3,3)
gap> gens := GeneratorsOfGroup(group1);
[ < a collineation: <cmat 3x3 over GF(3,1)>, F^0>,
 < a collineation: <cmat 3x3 over GF(3,1)>, F^0> ]
gap> group1_image := Group(List(gens,x->x^hom));
<projective collineation group with 2 generators>
gap> Order(group1_image);
5616
gap> group2 := ProjectivityGroup(pg2);
The FinInG projectivity group PGL(3,9)
gap> Order(group2);
42456960
gap> g := Random(group2);
< a collineation: <cmat 3x3 over GF(3,2)>, F^0>
gap> PreImageElm(hom,g);
#I <el> is not in the range of the intertwiner
fail

</pre></div>

<a id="X7BC7FCDC7D9E1A09" name="X7BC7FCDC7D9E1A09"></a>

<h5>10.4-3 Embedding of projective spaces by field reduction</h5>

We briefly describe the mathematics behind field reduction. For more details we refer to <a href="chapBib.html#biBLaVa">[LVdV13]</a>. Consider the fields GF(q) and GF(qt). The field GF(qt) is a t-dimensional vector space over GF(q). Hence, with respect to a chosen basis B for GF(qt) as a GF(q)-vector space, the bijection between the vector spaces V(n,qt) and V(tn,q) can be implemented. Consider the projective space PG(n-1,qt). The elements are represented by subspaces of V(n,qt). Clearly, a k dimensional subspace of V(n,qt) is also a kn-dimensional subspace of V(nt,q). This induces an embedding from PG(n-1,qt) into PG(nt-1,q). The embedding will be determined by the chosen basis of GF(qt) as a vector space over GF(q)

<a id="X7F84986879DD952A" name="X7F84986879DD952A"></a>

<h5>10.4-4 BlownUpSubspaceOfProjectiveSpace</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BlownUpSubspaceOfProjectiveSpace</code>( <var class="Arg">B</var>, <var class="Arg">subspace</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a subspace of a projective space

Let <var class="Arg">B</var> be a basis for the field GF(qt) as GF(q) vector space, and let <var class="Arg">subspace</var> be a k-1-dimensional subspace of PG(n-1,qt) represented by a k-dimensional subspace S of V(n,qt). This operation returns the kt-1-dimensional subspace of PG(nt-1,q) represented by blowing up S with respect to the base <var class="Arg">B</var>. This operation relies on the GAP operation <code class="file">BlownUpMat</code>. In the example, the effect of choosing a different basis is shown.

<div class="example"><pre>
gap> pg := PG(3,5^2);
ProjectiveSpace(3, 25)
gap> basis := Basis(AsVectorSpace(GF(5),GF(5^2)));
CanonicalBasis( GF(5^2) )
gap> line := Random(Lines(pg));
<a line in ProjectiveSpace(3, 25)>
gap> solid1 := BlownUpSubspaceOfProjectiveSpace(basis,line);
<a solid in ProjectiveSpace(7, 5)>
gap> BasisVectors(basis);
[ Z(5)^0, Z(5^2) ]
gap> basis := Basis(AsVectorSpace(GF(5),GF(5^2)),[Z(5),Z(5^2)^8]);
Basis( GF(5^2), [ Z(5), Z(5^2)^8 ] )
gap> solid2 := BlownUpSubspaceOfProjectiveSpace(basis,line);
<a solid in ProjectiveSpace(7, 5)>
gap> solid1 = solid2;
false

</pre></div>

<a id="X7D2D9AD287532FDB" name="X7D2D9AD287532FDB"></a>

<h5>10.4-5 NaturalEmbeddingByFieldReduction</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalEmbeddingByFieldReduction</code>( <var class="Arg">geom1</var>, <var class="Arg">f2</var>, <var class="Arg">B</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalEmbeddingByFieldReduction</code>( <var class="Arg">geom1</var>, <var class="Arg">f2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalEmbeddingByFieldReduction</code>( <var class="Arg">geom1</var>, <var class="Arg">geom2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalEmbeddingByFieldReduction</code>( <var class="Arg">geom1</var>, <var class="Arg">geom2</var>, <var class="Arg">B</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a geometry morphism

This operation comes in four flavours. For the first flavour, the argument <var class="Arg">geom1</var> is a projective space over a field GF(qt). The argument <var class="Arg">f2</var> is a subfield GF(q) of GF(qt). The argument B is a basis for GF(qt) as a GF(q)-vector space. When this argument is not given, a basis for GF(qt) over GF(q) is computed using <code class="file">Basis(Asvector space(GF(q),GF(q^t)))</code>. It is checked whether <var class="Arg">f2</var> is a subfield of the base field of <var class="Arg">geom1</var>. The third and fourth flavour are comparable, where now GF(q) is found as the base field of <var class="Arg">geom2</var>. In fact the arguments <var class="Arg">geom1</var> and <var class="Arg">geom2</var> are the projective spaces PG(n-1,qt) and PG(nt-1,q) respectively. As in the previous flavours, the argument <var class="Arg">B</var> is optional.

An intertwiner is always available for this geometry morphism, and has source the homography group of <var class="Arg">geom1</var> and as range a subgroup of the homography group of <var class="Arg">geom2</var> (or the projective space of the appropriate dimension over <var class="Arg">f2</var>. Notice in the example below the difference of a factor 2 in the orders of the group, which comes of course from restricting the homomorphism to the homography group, which differs a factor 2 from the collineation group of the projective line, that has an extra automorphism of order two, corresponding with the Frobenius automorphism.

<div class="example"><pre>
gap> pg1 := ProjectiveSpace(2,81);
ProjectiveSpace(2, 81)
gap> f2 := GF(9);
GF(3^2)
gap> em := NaturalEmbeddingByFieldReduction(pg1,f2);
<geometry morphism from <All elements of ProjectiveSpace(2,
81)> to <All elements of ProjectiveSpace(5, 9)>>
gap> f2 := GF(3);
GF(3)
gap> em := NaturalEmbeddingByFieldReduction(pg1,f2);
<geometry morphism from <All elements of ProjectiveSpace(2,
81)> to <All elements of ProjectiveSpace(11, 3)>>
gap> pg2 := ProjectiveSpace(11,3);
ProjectiveSpace(11, 3)
gap> em := NaturalEmbeddingByFieldReduction(pg1,pg2);
<geometry morphism from <All elements of ProjectiveSpace(2,
81)> to <All elements of ProjectiveSpace(11, 3)>>
gap> pg1 := PG(1,9);
ProjectiveSpace(1, 9)
gap> em := NaturalEmbeddingByFieldReduction(pg1,GF(3));
<geometry morphism from <All elements of ProjectiveSpace(1,
9)> to <All elements of ProjectiveSpace(3, 3)>>
gap> i := Intertwiner(em);
MappingByFunction( The FinInG projectivity group PGL(2,9), <projective colline
ation group of size 720 with
2 generators>, function( m ) ... end, function( m ) ... end )
gap> spread := List(Points(pg1),x->x^em);
[ <a line in ProjectiveSpace(3, 3)>, <a line in ProjectiveSpace(3, 3)>,
 <a line in ProjectiveSpace(3, 3)>, <a line in ProjectiveSpace(3, 3)>,
 <a line in ProjectiveSpace(3, 3)>, <a line in ProjectiveSpace(3, 3)>,
 <a line in ProjectiveSpace(3, 3)>, <a line in ProjectiveSpace(3, 3)>,
 <a line in ProjectiveSpace(3, 3)>, <a line in ProjectiveSpace(3, 3)> ]
gap> stab := Stabilizer(CollineationGroup(PG(3,3)),Set(spread),OnSets);
<projective collineation group of size 5760 with 3 generators>
gap> hom := HomographyGroup(pg1);
The FinInG projectivity group PGL(2,9)
gap> gens := GeneratorsOfGroup(hom);;
gap> group := Group(List(gens,x->x^i));
<projective collineation group with 2 generators>
gap> Order(group);
2880
gap> IsSubgroup(stab,group);
true

</pre></div>

<a id="X7C00DD48787B1EEE" name="X7C00DD48787B1EEE"></a>

<h4>10.5 Embeddings of polar spaces</h4>

<a id="X857976B679B8EC30" name="X857976B679B8EC30"></a>

<h5>10.5-1 NaturalEmbeddingBySubspace</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalEmbeddingBySubspace</code>( <var class="Arg">geom1</var>, <var class="Arg">geom2</var>, <var class="Arg">v</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a geometry morphism

The arguments <var class="Arg">geom1</var> and <var class="Arg">geom2</var> both polar spaces, and <var class="Arg">v</var> is an element of a projective space. This function returns a geometry morphism representing the natural embedding of <var class="Arg">geom1</var> into the intersection of <var class="Arg">geom2</var> and <var class="Arg">v</var>. Hence the intersection of <var class="Arg">geom2</var> and <var class="Arg">v</var> must induce a polar space of the same type as <var class="Arg">geom1</var> in <var class="Arg">v</var>. This operation performs all necessary checks. An Intertwiner is not implemented for this geometry morphism.

<div class="example"><pre>
gap> h1 := HermitianPolarSpace(2, 3^2);
H(2, 3^2)
gap> h2 := HermitianPolarSpace(3, 3^2);
H(3, 3^2)
gap> pg := AmbientSpace( h2 ); 
ProjectiveSpace(3, 9)
gap> pi := VectorSpaceToElement( pg, [[1,0,0,0],[0,1,0,0],[0,0,1,0]] * Z(9)^0 );
<a plane in ProjectiveSpace(3, 9)>
gap> em := NaturalEmbeddingBySubspace( h1, h2, pi );
<geometry morphism from <Elements of H(2, 3^2)> to <Elements of H(3, 3^2)>>
gap> ps1 := ParabolicQuadric(4,4);
Q(4, 4)
gap> ps2 := ParabolicQuadric(6,4);
Q(6, 4)
gap> pg := AmbientSpace( ps2 ); 
ProjectiveSpace(6, 4)
gap> pi := VectorSpaceToElement( pg, [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],
> [0,0,0,1,0,0,0],[0,0,0,0,1,0,0]] * Z(4)^0 );
<a proj. 4-space in ProjectiveSpace(6, 4)>
gap> em := NaturalEmbeddingBySubspace( ps1, ps2, pi );
<geometry morphism from <Elements of Q(4, 4)> to <Elements of Q(6, 4)>>
gap> List(Lines(ps1),x->x^em);
[ <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>,
 <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>,
 <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>,
 <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>,
 <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>,
 <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>,
 <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>,
 <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>,
 <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>,
 <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>,
 <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>,
 <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>,
 <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>,
 <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>,
 <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>,
 <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>,
 <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>,
 <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>,
 <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>,
 <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>,
 <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>,
 <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>,
 <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>,
 <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>,
 <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>,
 <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>,
 <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>,
 <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>,
 <a line in Q(6, 4)> ]

</pre></div>

<a id="X82510EF078F0806E" name="X82510EF078F0806E"></a>

<h5>10.5-2 NaturalEmbeddingBySubField</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalEmbeddingBySubField</code>( <var class="Arg">geom1</var>, <var class="Arg">geom2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalEmbeddingBySubField</code>( <var class="Arg">geom1</var>, <var class="Arg">geom2</var>, <var class="Arg">boolean</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a geometry morphism

The arguments <var class="Arg">geom1</var> and <var class="Arg">geom2</var> are projective or polar spaces with an underlying vector space of the same dimension and the base field L of <var class="Arg">geom2</var> is an extension of the base field K of <var class="Arg">geom1</var>. The form f determining <var class="Arg">geom1</var> also defines a form over L, and determines a polar space. By considering the underlying vector spaces determining the elements of <var class="Arg">geom1</var> over the extension field L, there is an obvious embedding of <var class="Arg">geom1</var> in the polar space over the extension field. Considering f over a field extension might change its type. The possible embeddings, where the polar spaces may be chosen up to equivalent form, are listed in the table below (see <a href="chapBib.html#biBKleidmanLiebeck">[KL90]</a>):

<div class="pcenter"><table class="GAPDocTable">
<caption class="GAPDocTable">Table: Subfield embeddings of polar spaces</caption>
<tr>
<td class="tdleft">Polar Space 1</td>
<td class="tdleft">Polar Space 2</td>
<td class="tdleft">Conditions</td>
</tr>
<tr>
<td class="tdleft">W(2n-1,q)</td>
<td class="tdleft">W(2n-1,qa)</td>
<td class="tdleft">--</td>
</tr>
<tr>
<td class="tdleft">W(2n-1,q)</td>
<td class="tdleft">H(2n-1,qa)</td>
<td class="tdleft">--</td>
</tr>
<tr>
<td class="tdleft">H(d,q2)</td>
<td class="tdleft">H(d,q2r)</td>
<td class="tdleft">r odd</td>
</tr>
<tr>
<td class="tdleft">Oε(d,q)</td>
<td class="tdleft">H(d,q2)</td>
<td class="tdleft">q odd</td>
</tr>
<tr>
<td class="tdleft">Oε(d,q)</td>
<td class="tdleft">Oε'(d,qr)</td>
<td class="tdleft">ε=(ε')r</td>
</tr>
</table> 
</div>

The geometry morphism also comes equipped with an intertwiner (see <code class="func">Intertwiner</code> (<a href="chap10.html#X86E0B2117DD536D2">10.1-2</a>)). The optional third argument <var class="Arg">boolean</var> can take either <code class="code">true</code> or <code class="code">false</code> as input, and then our operation will or will not compute the intertwiner accordingly. When set <code class="code">true</code>, the intertwiner will be computed if <code class="file">HasCollineationGroup(geom1)</code> is true. The user may wish that the intertwiner is not computed when embedding large polar spaces. The default (when calling the operation with two arguments) is set to <code class="code">true</code>.

<div class="example"><pre>
gap> w := SymplecticSpace(5, 3);
W(5, 3)
gap> h := HermitianPolarSpace(5, 3^2);
H(5, 3^2)
gap> em := NaturalEmbeddingBySubfield(w, h);
#I No intertwiner computed. <geom1> must have a collineation group computed
<geometry morphism from <Elements of W(5, 3)> to <Elements of H(5, 3^2)>>
gap> points := AsList(Points(w));;
gap> image := ImagesSet(em, points);;
gap> ForAll(image, x -> x in h);
true
gap> hq:=HyperbolicQuadric(3,4);
Q+(3, 4)
gap> eq:=EllipticQuadric(3,2);
Q-(3, 2)
gap> em:=NaturalEmbeddingBySubfield(eq,hq);
#I No intertwiner computed. <geom1> must have a collineation group computed
<geometry morphism from <Elements of Q-(3, 2)> to <Elements of Q+(3, 4)>>
gap> eqpts:=ImagesSet(em,AsList(Points(eq)));
[ <a point in Q+(3, 4)>, <a point in Q+(3, 4)>, <a point in Q+(3, 4)>,
 <a point in Q+(3, 4)>, <a point in Q+(3, 4)> ]

</pre></div>

<a id="X7823BA95797898CE" name="X7823BA95797898CE"></a>

<h5>10.5-3 Embedding of polar spaces by field reduction</h5>

Field reduction for polar spaces is somewhat more involved than for projective spaces, and we give a brief description. Let L be the field GF(qt) and let K be the field GF(q) Let P be a polar space over a field L. Let f be the form on the r dimensional vector space V over L determining P. Consider the trace map

T: L -> K: x ↦ xqt + xqt-1 + … + x.

Define for any α ∈ L the map

Tα: L: → K: x↦ Tα = T(α x).

Consider the rt dimensional vector space W over K. There is a bijective map Φ: V -> W and Tα • f • Φ -1 defines a quadratic or sesquilinear form (depending on α, and f being quadratic or sesquilinear) acting on W, and hence, if not singular or degenerate, inducing a polar space S over the finite field GF(q). An element of P can be mapped onto an element of S by simply blowing up P using field reduction for projective spaces. So the resulting polar space S is dependent on the original form f, the parameter α and the blowing up of elements by field reduction, the latter being dependent on the basis of L as a K vector space. FinInG provides two approaches. The first approach starts from P and the parameters K, α and a basis for L as K vector space. Then the resulting form Tα • f • Φ -1 is determined, and the associated polar space S will be the range of the embedding. Note that the resulting polar space will not necessarily be canonical. The second approach starts from two given polar spaces P and S. Based on this input, it is determined whether an embedding based on the above described principle is possible, and the necessary parameters are computed. The resulting embedding is a geometry morphism from P to S. Note that the polar spaces used as an argument may be freely chosen and are not required to be in the canonical form.

For more information on the embeddings by field reduction of polar spaces, including conditions on the parameter α, we refer to <a href="chapBib.html#biBGill">[Gil08]</a> and <a href="chapBib.html#biBLaVa">[LVdV13]</a>. The possible embeddings are listed in the following table.

<div class="pcenter"><table class="GAPDocTable">
<caption class="GAPDocTable">Table: Field reduction of polar spaces</caption>
<tr>
<td class="tdleft">Polar Space 1</td>
<td class="tdleft">Polar Space 2</td>
<td class="tdleft">Conditions</td>
</tr>
<tr>
<td class="tdleft">W(2n-1,qt)</td>
<td class="tdleft">W(2nt-1,q)</td>
<td class="tdleft">--</td>
</tr>
<tr>
<td class="tdleft">Q+(2n-1,qt)</td>
<td class="tdleft">Q+(2nt-1,q)</td>
<td class="tdleft">--</td>
</tr>
<tr>
<td class="tdleft">Q-(2n-1,qt)</td>
<td class="tdleft">Q-(2nt-1,q)</td>
<td class="tdleft">--</td>
</tr>
<tr>
<td class="tdleft">Q(2n,q2a+1)</td>
<td class="tdleft">Q((2a+1)(2n+1)-1,q)</td>
<td class="tdleft">q odd</td>
</tr>
<tr>
<td class="tdleft">Q(2n,q2a)</td>
<td class="tdleft">Q-(2a(2n+1)-1,q)</td>
<td class="tdleft">q ≡ 1 mod 4</td>
</tr>
<tr>
<td class="tdleft">Q(2n,q4a+2)</td>
<td class="tdleft">Q+((4a+2)(2n+1)-1,q)</td>
<td class="tdleft">q ≡ 3 mod 4</td>
</tr>
<tr>
<td class="tdleft">Q(2n,q4a)</td>
<td class="tdleft">Q-(4a(2n+1)-1,q)</td>
<td class="tdleft">q ≡ 3 mod 4</td>
</tr>
<tr>
<td class="tdleft">H(n,q2a+1)</td>
<td class="tdleft">H((2a+1)(n+1)-1,q)</td>
<td class="tdleft">q square</td>
</tr>
<tr>
<td class="tdleft">H(n,q2a)</td>
<td class="tdleft">W(2a(n+1)-1,q)</td>
<td class="tdleft">q even</td>
</tr>
<tr>
<td class="tdleft">H(2n,q2a)</td>
<td class="tdleft">Q-(2a(2n+1)-1,q)</td>
<td class="tdleft">q odd</td>
</tr>
<tr>
<td class="tdleft">H(2n+1,q2a)</td>
<td class="tdleft">Q+(2a(2n+2)-1,q)</td>
<td class="tdleft">q odd</td>
</tr>
</table> 
</div>

<a id="X7D2D9AD287532FDB" name="X7D2D9AD287532FDB"></a>

<h5>10.5-4 NaturalEmbeddingByFieldReduction</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalEmbeddingByFieldReduction</code>( <var class="Arg">ps1</var>, <var class="Arg">f2</var>, <var class="Arg">alpha</var>, <var class="Arg">basis</var>, <var class="Arg">bool</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalEmbeddingByFieldReduction</code>( <var class="Arg">ps1</var>, <var class="Arg">f2</var>, <var class="Arg">alpha</var>, <var class="Arg">basis</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalEmbeddingByFieldReduction</code>( <var class="Arg">ps1</var>, <var class="Arg">f2</var>, <var class="Arg">alpha</var>, <var class="Arg">bool</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalEmbeddingByFieldReduction</code>( <var class="Arg">ps1</var>, <var class="Arg">f2</var>, <var class="Arg">alpha</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalEmbeddingByFieldReduction</code>( <var class="Arg">ps1</var>, <var class="Arg">f2</var>, <var class="Arg">bool</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalEmbeddingByFieldReduction</code>( <var class="Arg">ps1</var>, <var class="Arg">f2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a geometry morphism

<var class="Arg">ps1</var> is a polar space over a field extension L of <var class="Arg">f2</var>, <var class="Arg">basis</var> is a basis for L over <var class="Arg">f2</var>, <var class="Arg">alpha</var> is a non-zero element of L. The version of <code class="file">NaturalEmbeddingByFieldReduction</code> implements the first approach as explained in <a href="chap10.html#X7823BA95797898CE">10.5-3</a>. When no argument <var class="Arg">basis</var> is given, a basis for L over <var class="Arg">f2</var> is computed using <code class="file">Basis(Asvector space(K,L))</code>. When no argument <var class="Arg">alpha</var> is given, <code class="file">One(f2)</code> is used as value for <var class="Arg">alpha</var>. When <var class="Arg">bool</var> is true or not given, an intertwiner is computed, when <var class="Arg">bool</var> is <var class="Arg">false</var>, no intertwiner is computed. This intertwiner has as its domain the isometry group of <var class="Arg">ps1</var>. The user may wish that the intertwiner is not computed when embedding large polar spaces. The default (when calling the operation with two arguments) is set to <code class="code">true</code>. In the first example, we construct a spread of maximal subspaces (solids) in a 7 dimensional symplectic space. We compute a subgroup of its stabilizer group using the intertwiner. In the second example, we construct a linear blocking set of the symplectic generalised quadrangle over GF(9) .

<div class="example"><pre>
gap> ps1 := SymplecticSpace(1,3^3);
W(1, 27)
gap> em := NaturalEmbeddingByFieldReduction(ps1,GF(3),true);
<geometry morphism from <Elements of W(1,
27)> to <Elements of <polar space in ProjectiveSpace(
5,GF(3)): -x1*y6-x2*y5-x3*y4-x3*y6+x4*y3+x5*y2+x6*y1+x6*y3=0 >>>
gap> ps2 := AmbientGeometry(Range(em));
<polar space in ProjectiveSpace(
5,GF(3)): -x1*y6-x2*y5-x3*y4-x3*y6+x4*y3+x5*y2+x6*y1+x6*y3=0 >
gap> spread := List(Points(ps1),x->x^em);;
gap> i := Intertwiner(em);
MappingByFunction( PSp(2,27), <projective collineation group of size
9828 with 2 generators>, function( m ) ... end, function( m ) ... end )
gap> coll := CollineationGroup(ps2);
#I Computing collineation group of canonical polar space...
<projective collineation group of size 9170703360 with 4 generators>
gap> stab := Group(ImagesSet(i,GeneratorsOfGroup(IsometryGroup(ps1))));
<projective collineation group with 2 generators>
gap> IsSubgroup(coll,stab);
true
gap> List(Orbit(stab,spread[1]),x->x in spread);
[ true, true, true, true, true, true, true, true, true, true, true, true,
 true, true, true, true, true, true, true, true, true, true, true, true,
 true, true, true, true ]

gap> ps1 := SymplecticSpace(3,9);
W(3, 9)
gap> em := NaturalEmbeddingByFieldReduction(ps1,GF(3),true);
<geometry morphism from <Elements of W(3,
9)> to <Elements of <polar space in ProjectiveSpace(
7
,GF(3)): -x1*y3+x1*y4+x2*y3+x3*y1-x3*y2-x4*y1-x5*y7+x5*y8+x6*y7+x7*y5-x7*y6-x8
*y5=0 >>>
gap> ps2 := AmbientGeometry(Range(em));
<polar space in ProjectiveSpace(
7
,GF(3)): -x1*y3+x1*y4+x2*y3+x3*y1-x3*y2-x4*y1-x5*y7+x5*y8+x6*y7+x7*y5-x7*y6-x8
*y5=0 >
gap> pg := AmbientSpace(ps2);
ProjectiveSpace(7, 3)
gap> spread := List(Points(ps1),x->x^em);;
gap> el := Random(ElementsOfIncidenceStructure(pg,5));
<a proj. 4-space in ProjectiveSpace(7, 3)>
gap> prebs := Filtered(spread,x->Meet(x,el) <> EmptySubspace(pg));;
gap> bs := List(prebs,x->PreImageElm(em,x));;
gap> Length(bs);
118
gap> lines := List(Lines(ps1));;
gap> Collected(List(lines,x->Length(Filtered(bs,y->y * x))));
[ [ 1, 702 ], [ 4, 117 ], [ 10, 1 ] ]

</pre></div>

<a id="X7D2D9AD287532FDB" name="X7D2D9AD287532FDB"></a>

<h5>10.5-5 NaturalEmbeddingByFieldReduction</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalEmbeddingByFieldReduction</code>( <var class="Arg">ps1</var>, <var class="Arg">ps2</var>, <var class="Arg">bool</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalEmbeddingByFieldReduction</code>( <var class="Arg">ps1</var>, <var class="Arg">ps2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a geometry morphism

If <var class="Arg">ps1</var> and <var class="Arg">ps2</var> are two polar spaces which are suitable for field reduction as listed in the table with possible embeddings in Section <a href="chap10.html#X7823BA95797898CE">10.5-3</a>, then this operation returns the corresponding embedding. An intertwiner is computed if the third argument <var class="Arg">bool</var> is true, or if there is no third argument. This intertwiner has as its domain the isometry group of <var class="Arg">ps1</var>. The example shows two cases where a spread is computed, including a subgroup of its stabiliser group using the intertwiner.

<div class="example"><pre>
gap> ps1 := SymplecticSpace(1,5^3);
W(1, 125)
gap> ps2 := SymplecticSpace(5,5);
W(5, 5)
gap> em := NaturalEmbeddingByFieldReduction(ps1,ps2);
#I These polar spaces are suitable for field reduction
<geometry morphism from <Elements of W(1, 125)> to <Elements of W(5, 5)>>
gap> pts := Points(ps1);
<points of W(1, 125)>
gap> spread := List(pts,x->x^em);;
gap> test := Union(List(spread,x->List(Points(x))));;
gap> Set(test)=Set(AsList(Points(ps2)));
true
gap> hom := Intertwiner(em);
MappingByFunction( PSp(2,125), <projective collineation group of size
976500 with 2 generators>, function( m ) ... end, function( m ) ... end )
gap> group := IsometryGroup(ps1);
PSp(2,125)
gap> Order(group);
976500
gap> gens := List(GeneratorsOfGroup(group),x->x^hom);
[ < a collineation: <cmat 6x6 over GF(5,1)>, F^0>,
 < a collineation: <cmat 6x6 over GF(5,1)>, F^0> ]
gap> group2 := Range(hom);
<projective collineation group of size 976500 with 2 generators>
gap> Order(group2);
976500
gap> biggroup := CollineationGroup(ps2);
PGammaSp(6,5)
gap> stab := FiningSetwiseStabiliser(biggroup,spread);
#I Computing adjusted stabilizer chain...
<projective collineation group with 7 generators>
gap> time;
6907
gap> Order(stab);
5859000
gap> ps1 := HermitianPolarSpace(2,7^2);
H(2, 7^2)
gap> ps2 := EllipticQuadric(5,7);
Q-(5, 7)
gap> em := NaturalEmbeddingByFieldReduction(ps1,ps2);
#I These polar spaces are suitable for field reduction
<geometry morphism from <Elements of H(2, 7^2)> to <Elements of Q-(5, 7)>>
gap> pts := Points(ps1);
<points of H(2, 7^2)>
gap> spread := List(pts,x->x^em);;
gap> test := Union(List(spread,x->List(Points(x))));;
gap> Set(test)=Set(AsList(Points(ps2)));
true
gap> hom := Intertwiner(em);
MappingByFunction( PGU(3,7^2), <projective collineation group of size
5663616 with 2 generators>, function( m ) ... end, function( m ) ... end )
gap> group := IsometryGroup(ps1);
PGU(3,7^2)
gap> Order(group);
5663616
gap> gens := List(GeneratorsOfGroup(group),x->x^hom);
[ < a collineation: <cmat 6x6 over GF(7,1)>, F^0>,
 < a collineation: <cmat 6x6 over GF(7,1)>, F^0> ]
gap> group2 := Range(hom);
<projective collineation group of size 5663616 with 2 generators>
gap> Order(group2);
5663616
gap> biggroup := CollineationGroup(ps2);
PDeltaO-(6,7)
gap> stab := FiningSetwiseStabiliser(biggroup,spread);
#I Computing adjusted stabilizer chain...
<projective collineation group with 10 generators>
gap> time;
3438
gap> Order(stab);
90617856

</pre></div>

<a id="X81FAC1DE7C4B1972" name="X81FAC1DE7C4B1972"></a>

<h4>10.6 Projections</h4>

<a id="X7B9E8AB682EC07C2" name="X7B9E8AB682EC07C2"></a>

<h5>10.6-1 NaturalProjectionBySubspace</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalProjectionBySubspace</code>( <var class="Arg">ps</var>, <var class="Arg">v</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a geometry morphism

The argument <var class="Arg">ps</var> is a projective or polar space, and <var class="Arg">v</var> is a subspace of <var class="Arg">ps</var>. In the case that <var class="Arg">ps</var> is a projective space, the geometry of subspaces containing <var class="Arg">v</var> is a projective space of lower dimension over the same base field, and this operation returns the corresponding geometry morphism. In the case that <var class="Arg">ps</var> is a polar space, the geometry of elements pf <var class="Arg">ps</var> containing <var class="Arg">v</var> is a polar space of lower rank and of the same type over the same base field, and this operation returns the corresponding geometry morphism. It is checked whether <var class="Arg">v</var> is a subspace of <var class="Arg">ps</var>, and whether the input of the function and preimage of the returned geometry morphism is valid or not. There is a shorthand for this operation which is basically an overload of the quotient operation. So, for example, <code class="code">ps / v</code> achieves the same thing as <code class="code">AmbientGeometry(Range(NaturalProjectionBySubspace(ps, v)))</code>. An intertwiner is not available for this geometry morphism.

<div class="example"><pre>
gap> ps := HyperbolicQuadric(5,3);
Q+(5, 3)
gap> x := Random(Points(ps));;
gap> planes_on_x := AsList( Planes(x) );
[ <a plane in Q+(5, 3)>, <a plane in Q+(5, 3)>, <a plane in Q+(5, 3)>,
 <a plane in Q+(5, 3)>, <a plane in Q+(5, 3)>, <a plane in Q+(5, 3)>,
 <a plane in Q+(5, 3)>, <a plane in Q+(5, 3)> ]
gap> proj := NaturalProjectionBySubspace(ps, x);
<geometry morphism from <Elements of Q+(5,
3)> to <Elements of <polar space in ProjectiveSpace(
3,GF(3)): x_1*x_2+x_3*x_4=0 >>>
gap> image := ImagesSet(proj, planes_on_x);
[ <a line in Q+(3, 3): x_1*x_2+x_3*x_4=0>,
 <a line in Q+(3, 3): x_1*x_2+x_3*x_4=0>,
 <a line in Q+(3, 3): x_1*x_2+x_3*x_4=0>,
 <a line in Q+(3, 3): x_1*x_2+x_3*x_4=0>,
 <a line in Q+(3, 3): x_1*x_2+x_3*x_4=0>,
 <a line in Q+(3, 3): x_1*x_2+x_3*x_4=0>,
 <a line in Q+(3, 3): x_1*x_2+x_3*x_4=0>,
 <a line in Q+(3, 3): x_1*x_2+x_3*x_4=0> ]

</pre></div>

<a id="X7952EE1A80D53825" name="X7952EE1A80D53825"></a>

<h4>10.7 Projective completion</h4>

<a id="X7E23CA137AEFD1D2" name="X7E23CA137AEFD1D2"></a>

<h5>10.7-1 ProjectiveCompletion</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ProjectiveCompletion</code>( <var class="Arg">as</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a geometry morphism

The argument <var class="Arg">as</var> is an affine space. This operation returns an embedding of <var class="Arg">as</var> into the projective space <var class="Arg">ps</var> of the same dimension, and over the same field. For example, the point (x, y, z) is mapped onto the projective point with homogeneous coordinates (1, x, y, z). An intertwiner is unnecessary, <code class="file">CollineationGroup(as)</code> is a subgroup of <code class="file">CollineationGroup(ps)</code>.

<div class="example"><pre>
gap> as := AffineSpace(3,5);
AG(3, 5)
gap> map := ProjectiveCompletion(as);
<geometry morphism from <Elements of AG(3,
5)> to <All elements of ProjectiveSpace(3, 5)>>
gap> p := Random( Points(as) );
<a point in AG(3, 5)>
gap> p^map;
<a point in ProjectiveSpace(3, 5)>

</pre></div>

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