  
  [1X10 [33X[0;0YGeometry Morphisms[133X[101X
  
  [33X[0;0YHere  we  describe  what  is  meant by a [13Xgeometry morphism[113X in [5XFinInG[105X and the
  various  operations  and  tools  available to the user. When using groups in
  [5XGAP[105X,  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  [5XGAP[105X,  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.[133X
  
  [33X[0;0YSuppose that [22XS[122X and [22XS'[122X are two incidence geometries. A [13Xgeometry morphism[113X from
  [22XS[122X  to [22XS'[122X is defined to be a map from the elements of [22XS[122X to the elements of [22XS'[122X
  which  preserves  incidence and induces a function from the type set of [22XS[122X to
  the  type  set  of  [22XS'[122X.  For  instance, a correlation and a collineation are
  examples  of  geometry  morphisms,  but  they  have  been dealt with in more
  specific ways in [5XFinInG[105X. 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.[133X
  
  [33X[0;0YAs  a  geometry morphism from [22XS[122X to [22XS'[122X preserves incidence, it also preserves
  the symmetry, and hence it induces also a map from the collineation group of
  [22XS[122X  into  the  collineation  group  of  [22XS'[122X.  Such  a  map  will  be called an
  [13XIntertwiner[113X,  and  [5XFinInG[105X  can  provide  these maps for some of the geometry
  morphisms.[133X
  
  [33X[0;0YNote 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 [14XC[114X.[133X
  
  
  [1X10.1 [33X[0;0YGeometry morphisms in FinInG[133X[101X
  
  [1X10.1-1 IsGeometryMorphism[101X
  
  [33X[1;0Y[29X[2XIsGeometryMorphism[102X [32X family[133X
  
  [33X[0;0YThe  category [10XIsGeometryMorphism[110X represents a special object in [5XFinInG[105X which
  carries  attributes and the given element map. The element map is given as a
  [10XIsGeneralMapping[110X, and so has a source and range.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XShowImpliedFilters(IsGeometryMorphism);[127X[104X
    [4X[28XImplies:[128X[104X
    [4X[28X   IsGeneralMapping[128X[104X
    [4X[28X   IsTotal[128X[104X
    [4X[28X   Tester(IsTotal)[128X[104X
    [4X[28X   IsSingleValued[128X[104X
    [4X[28X   Tester(IsSingleValued)[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  usual  operations  of [10XImageElm[110X, and [10XPreImageElm[110X, have methods installed
  for geometry morphisms, as well as the overload operator [10X\^[110X.[133X
  
  [1X10.1-2 Intertwiner[101X
  
  [33X[1;0Y[29X[2XIntertwiner[102X( [3Xf[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya group homomorphism[133X
  
  [33X[0;0YThe  argument  [3Xf[103X  is a geometry morphism. If [3Xf[103X comes equipped with a natural
  intertwiner   from  an  automorphism  group  of  the  source  of  [3Xf[103X  to  the
  automorphism  group  to  the image of [3Xf[103X, 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 [22Xf[122X from [22XS[122X to [22XS'[122X, an intertwiner [22Xϕ[122X is a map
  from  the automorphism group of [22XS[122X to the automorphism group of [22XS'[122X, such that
  for  every  element  [22Xp[122X  of [22XS[122X and every automorphism [22Xg[122X of [22XS[122X, we have [22Xf(p^g) =
  f(p)^ϕ(g).[122X[133X
  
  [33X[0;0YThere  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 [11XIntertwiner[111X available. Here is a simple example of the intertwiner
  for   the  isomorphism  of  two  polar  spaces  (see  [2XIsomorphismPolarSpaces[102X
  ([14X10.2-1[114X)). The source of the homomorphism is dependent on the geometry.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xform := BilinearFormByMatrix( IdentityMat(3,GF(3)), GF(3) );[127X[104X
    [4X[28X< bilinear form >[128X[104X
    [4X[25Xgap>[125X [27Xps := PolarSpace(form);[127X[104X
    [4X[28X<polar space in ProjectiveSpace(2,GF(3)): x_1^2+x_2^2+x_3^2=0 >[128X[104X
    [4X[25Xgap>[125X [27Xpq := ParabolicQuadric(2,3);[127X[104X
    [4X[28Xstandard Q(2, 3)[128X[104X
    [4X[25Xgap>[125X [27Xiso := IsomorphismPolarSpaces(ps, pq);[127X[104X
    [4X[28X#I  Computing nice monomorphism...[128X[104X
    [4X[28X<geometry morphism from <Elements of <polar space in ProjectiveSpace(2,GF([128X[104X
    [4X[28X3)): x_1^2+x_2^2+x_3^2=0 >> to <Elements of standard Q(2, 3)>>[128X[104X
    [4X[25Xgap>[125X [27XKnownAttributesOfObject(iso);[127X[104X
    [4X[28X[ "Range", "Source", "Intertwiner" ][128X[104X
    [4X[25Xgap>[125X [27Xhom := Intertwiner(iso);[127X[104X
    [4X[28XMappingByFunction( <projective semilinear group with [128X[104X
    [4X[28X3 generators>, PGammaO(3,3), function( y ) ... end, function( x ) ... end )[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  
  [1X10.2 [33X[0;0YType preserving bijective geometry morphisms[133X[101X
  
  [33X[0;0YAn  important  class  of  geometry  morphisms in [5XFinInG[105X are the isomorphisms
  between  polar  spaces  of  the  same  kind  that  are induced by coordinate
  transformations.[133X
  
  [1X10.2-1 IsomorphismPolarSpaces[101X
  
  [33X[1;0Y[29X[2XIsomorphismPolarSpaces[102X( [3Xps1[103X, [3Xps2[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsomorphismPolarSpaces[102X( [3Xps1[103X, [3Xps2[103X, [3Xboolean[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya geometry morphism[133X
  
  [33X[0;0YThe arguments [3Xps1[103X and [3Xps2[103X are [13Xequivalent[113X polar spaces, i.e. up to coordinate
  transformation, the underlying sesquilinear or quadratic form determines the
  same  polar  space,  or, [3Xps1[103X is a parabolic quadric over a finite field [22Xf[122X of
  even  characteristic in dimension [22X2n[122X and [3Xps2[103X is a symplectic space over [22Xf[122X in
  dimension  [22X2n-1[122X,  then this operation returns a geometry isomorphism between
  them.  The  optional third argument [3Xboolean[103X can take either [10Xtrue[110X or [10Xfalse[110X 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 [10Xtrue[110X, and in this case, if at least one of [3Xps1[103X
  or  [3Xps2[103X  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  [3Xps1[103X  onto  the  collineation group of [3Xps2[103X (see also
  [2XIntertwiner[102X ([14X10.1-2[114X)).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmat1 := IdentityMat(6,GF(5));[127X[104X
    [4X[28X< mutable compressed matrix 6x6 over GF(5) >[128X[104X
    [4X[25Xgap>[125X [27Xform1 := BilinearFormByMatrix(mat1,GF(5));[127X[104X
    [4X[28X< bilinear form >[128X[104X
    [4X[25Xgap>[125X [27Xps1 := PolarSpace(form1);[127X[104X
    [4X[28X<polar space in ProjectiveSpace([128X[104X
    [4X[28X5,GF(5)): x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2=0 >[128X[104X
    [4X[25Xgap>[125X [27Xmat2 := [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[127X[104X
    [4X[25X>[125X [27X[0,0,0,0,0,0],[0,0,0,0,0,0],[0,0,0,0,0,0]]*Z(5)^0;[127X[104X
    [4X[28X[ [ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), Z(5)^0 ], [128X[104X
    [4X[28X  [ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), Z(5)^0, 0*Z(5) ], [128X[104X
    [4X[28X  [ 0*Z(5), 0*Z(5), 0*Z(5), Z(5)^0, 0*Z(5), 0*Z(5) ], [128X[104X
    [4X[28X  [ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5) ], [128X[104X
    [4X[28X  [ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5) ], [128X[104X
    [4X[28X  [ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5) ] ][128X[104X
    [4X[25Xgap>[125X [27Xform2 := QuadraticFormByMatrix(mat2,GF(5));[127X[104X
    [4X[28X< quadratic form >[128X[104X
    [4X[25Xgap>[125X [27Xps2 := PolarSpace(form2);[127X[104X
    [4X[28X<polar space in ProjectiveSpace(5,GF(5)): x_1*x_6+x_2*x_5+x_3*x_4=0 >[128X[104X
    [4X[25Xgap>[125X [27Xiso := IsomorphismPolarSpaces(ps1,ps2,true);[127X[104X
    [4X[28X#I  No intertwiner computed. One of the polar spaces must have a collineation group computed[128X[104X
    [4X[28X<geometry morphism from <Elements of Q+(5, [128X[104X
    [4X[28X5): x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2=0> to <Elements of Q+(5, [128X[104X
    [4X[28X5): x_1*x_6+x_2*x_5+x_3*x_4=0>>[128X[104X
    [4X[25Xgap>[125X [27XCollineationGroup(ps1);[127X[104X
    [4X[28X#I  Computing collineation group of canonical polar space...[128X[104X
    [4X[28X<projective collineation group of size 58032000000 with 4 generators>[128X[104X
    [4X[25Xgap>[125X [27XCollineationGroup(ps2);[127X[104X
    [4X[28X#I  Computing collineation group of canonical polar space...[128X[104X
    [4X[28X<projective collineation group of size 58032000000 with 4 generators>[128X[104X
    [4X[25Xgap>[125X [27Xiso := IsomorphismPolarSpaces(ps1,ps2,true);[127X[104X
    [4X[28X<geometry morphism from <Elements of Q+(5, [128X[104X
    [4X[28X5): x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2=0> to <Elements of Q+(5, [128X[104X
    [4X[28X5): x_1*x_6+x_2*x_5+x_3*x_4=0>>[128X[104X
    [4X[25Xgap>[125X [27Xhom := Intertwiner( iso );[127X[104X
    [4X[28XMappingByFunction( <projective collineation group of size 58032000000 with [128X[104X
    [4X[28X4 generators>, <projective collineation group of size 58032000000 with [128X[104X
    [4X[28X4 generators>, function( y ) ... end, function( x ) ... end )[128X[104X
    [4X[25Xgap>[125X [27Xps1 := ParabolicQuadric(6,8);[127X[104X
    [4X[28XQ(6, 8)[128X[104X
    [4X[25Xgap>[125X [27Xps2 := SymplecticSpace(5,8);[127X[104X
    [4X[28XW(5, 8)[128X[104X
    [4X[25Xgap>[125X [27Xem := IsomorphismPolarSpaces(ps1,ps2);[127X[104X
    [4X[28X#I  Have 36171 points.[128X[104X
    [4X[28X#I  Have 37381 points in new orbit.[128X[104X
    [4X[28X#I  Have 36171 points.[128X[104X
    [4X[28X#I  Have 37388 points in new orbit.[128X[104X
    [4X[28X<geometry morphism from <Elements of Q(6, 8)> to <Elements of W(5, 8)>>[128X[104X
    [4X[25Xgap>[125X [27Xhom := Intertwiner(em);[127X[104X
    [4X[28XMappingByFunction( PGammaO(7,8), <projective collineation group of size [128X[104X
    [4X[28X27231016821530296320 with [128X[104X
    [4X[28X3 generators>, function( el ) ... end, function( el ) ... end )[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  
  [1X10.3 [33X[0;0YKlein correspondence and derived dualities[133X[101X
  
  [33X[0;0YThe Klein correspondence is a well known geometry morphism from the lines of
  [22XPG(3,q)[122X  to the points of a hyperbolic quadric in [22XPG(5,q)[122X. This morphism and
  some  derived  morphisms  are  provided in [5XFinInG[105X. The bare essential of the
  Klein correspondence is the so-called Plücker map.[133X
  
  [1X10.3-1 PluckerCoordinates[101X
  
  [33X[1;0Y[29X[2XPluckerCoordinates[102X( [3Xline[103X ) [32X operation[133X
  
  [33X[0;0YThis  operation  takes a line of [22XPG(3,q)[122X 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 [11Xvector spaceToElement[111X, and represents a
  point    of    the    hyperbolic    quadric   in   [22XPG(5,q)[122X   with   equation
  [22XX_0X_5+X_1X_4+X_2X_3 = 0[122X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpg := PG(3,169);[127X[104X
    [4X[28XProjectiveSpace(3, 169)[128X[104X
    [4X[25Xgap>[125X [27Xl := Random(Lines(pg));[127X[104X
    [4X[28X<a line in ProjectiveSpace(3, 169)>[128X[104X
    [4X[25Xgap>[125X [27Xvec := PluckerCoordinates(l);[127X[104X
    [4X[28X[ Z(13)^0, Z(13^2)^138, Z(13^2)^93, Z(13^2)^53, Z(13^2)^71, Z(13^2)^106 ][128X[104X
    [4X[25Xgap>[125X [27Xmat := [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[127X[104X
    [4X[25X>[125X [27X[0,0,0,0,0,0],[0,0,0,0,0,0],[0,0,0,0,0,0]]*Z(13)^0;[127X[104X
    [4X[28X[ [ 0*Z(13), 0*Z(13), 0*Z(13), 0*Z(13), 0*Z(13), Z(13)^0 ], [128X[104X
    [4X[28X  [ 0*Z(13), 0*Z(13), 0*Z(13), 0*Z(13), Z(13)^0, 0*Z(13) ], [128X[104X
    [4X[28X  [ 0*Z(13), 0*Z(13), 0*Z(13), Z(13)^0, 0*Z(13), 0*Z(13) ], [128X[104X
    [4X[28X  [ 0*Z(13), 0*Z(13), 0*Z(13), 0*Z(13), 0*Z(13), 0*Z(13) ], [128X[104X
    [4X[28X  [ 0*Z(13), 0*Z(13), 0*Z(13), 0*Z(13), 0*Z(13), 0*Z(13) ], [128X[104X
    [4X[28X  [ 0*Z(13), 0*Z(13), 0*Z(13), 0*Z(13), 0*Z(13), 0*Z(13) ] ][128X[104X
    [4X[25Xgap>[125X [27Xform := QuadraticFormByMatrix(mat,GF(169));[127X[104X
    [4X[28X< quadratic form >[128X[104X
    [4X[25Xgap>[125X [27Xklein := PolarSpace(form);[127X[104X
    [4X[28X<polar space in ProjectiveSpace(5,GF(13^2)): x_1*x_6+x_2*x_5+x_3*x_4=0 >[128X[104X
    [4X[25Xgap>[125X [27XVectorSpaceToElement(klein,vec);[127X[104X
    [4X[28X<a point in Q+(5, 169): x_1*x_6+x_2*x_5+x_3*x_4=0>[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X10.3-2 KleinCorrespondence[101X
  
  [33X[1;0Y[29X[2XKleinCorrespondence[102X( [3Xf[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XKleinCorrespondence[102X( [3Xf[103X, [3Xboolean[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XKleinCorrespondence[102X( [3Xq[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XKleinCorrespondence[102X( [3Xq[103X, [3Xboolean[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya geometry morphism[133X
  
  [33X[0;0YThe argument [3Xf[103X is a finite field, the argument [3Xq[103X is a prime power. The first
  an the third version use [9Xtrue[109X as value for [3Xboolean[103X. When using [9Xtrue[109X as value
  for  the boolean, the intertwiner is computed. This variant of the operation
  [11XKleinCorrespondence[111X  has  always  as  ambient  geometry  of  its  range  the
  hyperbolic  quadric  [22XQ^+(5,q)[122X  with  equation  [22XX_0X_5+X_1X_4+X_2X_3  = 0[122X The
  returned geometry morphism has the lines of [22XPG(3,q)[122X as source and the points
  of [22XQ^+(5,q)[122X as range.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xk := KleinCorrespondence( 9 );[127X[104X
    [4X[28X<geometry morphism from <lines of ProjectiveSpace(3, 9)> to <points of Q+(5, [128X[104X
    [4X[28X9): x_1*x_6+x_2*x_5+x_3*x_4=0>>[128X[104X
    [4X[25Xgap>[125X [27XIntertwiner(k);[127X[104X
    [4X[28XMappingByFunction( The FinInG collineation group PGammaL(4,9), <projective col[128X[104X
    [4X[28Xlineation group with [128X[104X
    [4X[28X3 generators>, function( g ) ... end, function( g ) ... end )[128X[104X
    [4X[25Xgap>[125X [27Xpg := ProjectiveSpace(3, 9);[127X[104X
    [4X[28XProjectiveSpace(3, 9)[128X[104X
    [4X[25Xgap>[125X [27XAmbientGeometry(Range(k));[127X[104X
    [4X[28XQ+(5, 9): x_1*x_6+x_2*x_5+x_3*x_4=0[128X[104X
    [4X[25Xgap>[125X [27Xl := Random( Lines(pg) );[127X[104X
    [4X[28X<a line in ProjectiveSpace(3, 9)>[128X[104X
    [4X[25Xgap>[125X [27Xl^k;[127X[104X
    [4X[28X<a point in Q+(5, 9): x_1*x_6+x_2*x_5+x_3*x_4=0>[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X10.3-3 KleinCorrespondence[101X
  
  [33X[1;0Y[29X[2XKleinCorrespondence[102X( [3Xquadric[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XKleinCorrespondence[102X( [3Xquadric[103X, [3Xboolean[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya geometry morphism[133X
  
  [33X[0;0YThe  argument  [3Xquadric[103X is a hyperbolic quadric in a 5 dimensional projective
  space.  If [3Xboolean[103X is [9Xtrue[109X or not given, this operation returns the geometry
  morphism  equipped  with  an intertwiner. The returned geometry morphism has
  the lines of [22XPG(3,q)[122X as source and the points of [22XQ^+(5,q)[122X as range.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xquadric := HyperbolicQuadric(5,3);                                         [127X[104X
    [4X[28XQ+(5, 3)[128X[104X
    [4X[25Xgap>[125X [27Xk := KleinCorrespondence( quadric );[127X[104X
    [4X[28X<geometry morphism from <lines of ProjectiveSpace(3, 3)> to <points of Q+(5, [128X[104X
    [4X[28X3)>>[128X[104X
    [4X[25Xgap>[125X [27Xpg := ProjectiveSpace(3, 3);[127X[104X
    [4X[28XProjectiveSpace(3, 3)[128X[104X
    [4X[25Xgap>[125X [27Xl := Random( Lines(pg) );[127X[104X
    [4X[28X<a line in ProjectiveSpace(3, 3)>[128X[104X
    [4X[25Xgap>[125X [27Xl^k;[127X[104X
    [4X[28X<a point in Q+(5, 3)>[128X[104X
    [4X[25Xgap>[125X [27Xid := IdentityMat(6,GF(13));[127X[104X
    [4X[28X< mutable compressed matrix 6x6 over GF(13) >[128X[104X
    [4X[25Xgap>[125X [27Xform := QuadraticFormByMatrix(id,GF(13));[127X[104X
    [4X[28X< quadratic form >[128X[104X
    [4X[25Xgap>[125X [27Xquadric := PolarSpace(form);[127X[104X
    [4X[28X<polar space in ProjectiveSpace([128X[104X
    [4X[28X5,GF(13)): x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2=0 >[128X[104X
    [4X[25Xgap>[125X [27Xk := KleinCorrespondence( quadric );[127X[104X
    [4X[28X<geometry morphism from <lines of ProjectiveSpace(3, 13)> to <points of Q+(5, [128X[104X
    [4X[28X13): x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2=0>>[128X[104X
    [4X[25Xgap>[125X [27Xpg := AmbientGeometry(Source(k));[127X[104X
    [4X[28XProjectiveSpace(3, 13)[128X[104X
    [4X[25Xgap>[125X [27Xl := Random(Lines(pg));[127X[104X
    [4X[28X<a line in ProjectiveSpace(3, 13)>[128X[104X
    [4X[25Xgap>[125X [27Xl^k;[127X[104X
    [4X[28X<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>[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X10.3-4 KleinCorrespondenceExtended[101X
  
  [33X[1;0Y[29X[2XKleinCorrespondenceExtended[102X( [3Xquadric[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XKleinCorrespondenceExtended[102X( [3Xquadric[103X, [3Xboolean[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya geometry morphism[133X
  
  [33X[0;0YThe  argument  [3Xquadric[103X is a hyperbolic quadric in a 5 dimensional projective
  space.  If [3Xboolean[103X is [9Xtrue[109X or not given, this operation returns the geometry
  morphism  equipped  with  an intertwiner. The returned geometry morphism has
  all  the elements of [22XPG(3,q)[122X as source (not just the lines) and the elements
  of  [22XQ^+(5,q)[122X  as  range,  hence  this  operation  is  a kind of extension of
  [11XKleinCorrespondence[111X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xps := HyperbolicQuadric(5,7);[127X[104X
    [4X[28XQ+(5, 7)[128X[104X
    [4X[25Xgap>[125X [27Xem := KleinCorrespondenceExtended(ps);[127X[104X
    [4X[28X<geometry morphism from <All elements of ProjectiveSpace(3, [128X[104X
    [4X[28X7)> to <Elements of Q+(5, 7)>>[128X[104X
    [4X[25Xgap>[125X [27Xhom := Intertwiner(em);[127X[104X
    [4X[28XMappingByFunction( The FinInG collineation group PGL(4,7), <projective colline[128X[104X
    [4X[28Xation group with 2 generators>, function( g ) ... end, function( g ) ... end )[128X[104X
    [4X[25Xgap>[125X [27Xmat := [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[127X[104X
    [4X[25X>[125X [27X	[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]*Z(7)^0;[127X[104X
    [4X[28X[ [ 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7), Z(7)^0 ], [128X[104X
    [4X[28X  [ 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7), Z(7)^0, 0*Z(7) ], [128X[104X
    [4X[28X  [ 0*Z(7), 0*Z(7), 0*Z(7), Z(7)^0, 0*Z(7), 0*Z(7) ], [128X[104X
    [4X[28X  [ 0*Z(7), 0*Z(7), Z(7)^0, 0*Z(7), 0*Z(7), 0*Z(7) ], [128X[104X
    [4X[28X  [ 0*Z(7), Z(7)^0, 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7) ], [128X[104X
    [4X[28X  [ Z(7)^0, 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7) ] ][128X[104X
    [4X[25Xgap>[125X [27Xg := Projectivity(mat,GF(7));[127X[104X
    [4X[28X< a collineation: <cmat 6x6 over GF(7,1)>, F^0>[128X[104X
    [4X[25Xgap>[125X [27Xg in CollineationGroup(ps);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XPreImageElm(hom,g);[127X[104X
    [4X[28X#I  <el> is not inducing a collineation of PG(3,q)[128X[104X
    [4X[28Xfail[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIt  is  well  known  that  the  classical generalised quadrangles [22XW(3,q)[122X and
  [22XQ(4,q)[122X  are  dual  incidence  structures,  the  same holds for the classical
  generalised  quadrangles  [22XQ^-(5,q)[122X  and  [22XH(3,q^2)[122X.  Essentially,  these dual
  dualities are based on the Klein correspondence, and are implemented through
  the operation [11XNaturalDuality[111X, this operation will return a geometry morphism
  with       [10XElementsOfIncidenceStructure(gq1)[110X       as       source       and
  [10XElementsOfIncidenceStructure(gq2)[110X as range, in other words, it is a geometry
  morphism  from  all  the  elements  of  [3Xgq1[103X  onto  all  the elements of [3Xgq2[103X,
  preserving the incidence, and swapping the types.[133X
  
  [1X10.3-5 NaturalDuality[101X
  
  [33X[1;0Y[29X[2XNaturalDuality[102X( [3Xgq1[103X, [3Xgq2[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XNaturalDuality[102X( [3Xgq1[103X, [3Xgq2[103X, [3Xboolean[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XNaturalDuality[102X( [3Xgq[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XNaturalDuality[102X( [3Xgq[103X, [3Xboolean[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya geometry morphism[133X
  
  [33X[0;0YThe  operation  allows  the  construction  of the duality between [22XW(3,q)[122X and
  [22XQ(4,q)[122X,  respectively  [22XQ^-(5,q)[122X,  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 [22X3[122X), 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 [5XFinInG[105X.[133X
  
  [33X[0;0YThe  third  version  requires  only  one  argument,  either  [22XW(3,q)[122X, [22XQ(4,q)[122X,
  [22XQ^-(5,q)[122X,  or  [22XH(3,q^2)[122X,  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.[133X
  
  [33X[0;0YThe  first and third version without a boolean as argument will, if possible
  return  a  geometry morphism equipped with an intertwiner. Using the boolean
  argument  [3Xfalse[103X will return a geometry morphism that is not equipped with an
  intertwiner.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xw := SymplecticSpace(3,5);[127X[104X
    [4X[28XW(3, 5)[128X[104X
    [4X[25Xgap>[125X [27Xlines:=AsList(Lines(w));;[127X[104X
    [4X[25Xgap>[125X [27Xduality := NaturalDuality(w);[127X[104X
    [4X[28X<geometry morphism from <Elements of W(3, 5)> to <Elements of Q(4, 5)>>[128X[104X
    [4X[25Xgap>[125X [27Xl:=lines[1];[127X[104X
    [4X[28X<a line in W(3, 5)>[128X[104X
    [4X[25Xgap>[125X [27Xl^duality;[127X[104X
    [4X[28X<a point in Q(4, 5)>[128X[104X
    [4X[25Xgap>[125X [27XPreImageElm(duality,last);[127X[104X
    [4X[28X<a line in W(3, 5)>[128X[104X
    [4X[25Xgap>[125X [27Xhom := Intertwiner(duality);[127X[104X
    [4X[28XMappingByFunction( PGammaSp(4,5), <projective collineation group of size [128X[104X
    [4X[28X9360000 with 4 generators>, function( g ) ... end, function( g ) ... end )[128X[104X
    [4X[25Xgap>[125X [27Xq := 5;[127X[104X
    [4X[28X5[128X[104X
    [4X[25Xgap>[125X [27Xq5q := EllipticQuadric(5,q);[127X[104X
    [4X[28XQ-(5, 5)[128X[104X
    [4X[25Xgap>[125X [27Xmat := [[0,1,0,0],[1,0,0,0],[0,0,0,Z(q)],[0,0,Z(q),0]]*Z(q)^0;[127X[104X
    [4X[28X[ [ 0*Z(5), Z(5)^0, 0*Z(5), 0*Z(5) ], [ Z(5)^0, 0*Z(5), 0*Z(5), 0*Z(5) ], [128X[104X
    [4X[28X  [ 0*Z(5), 0*Z(5), 0*Z(5), Z(5) ], [ 0*Z(5), 0*Z(5), Z(5), 0*Z(5) ] ][128X[104X
    [4X[25Xgap>[125X [27Xhform := HermitianFormByMatrix(mat,GF(q^2));[127X[104X
    [4X[28X< hermitian form >[128X[104X
    [4X[25Xgap>[125X [27Xherm := PolarSpace(hform);[127X[104X
    [4X[28X<polar space in ProjectiveSpace([128X[104X
    [4X[28X3,GF(5^2)): x1^5*x2+x1*x2^5+Z(5)*x3^5*x4+Z(5)*x3*x4^5=0 >[128X[104X
    [4X[25Xgap>[125X [27Xduality := NaturalDuality(q5q,herm,true);[127X[104X
    [4X[28X<geometry morphism from <Elements of Q-(5, 5)> to <Elements of H(3, [128X[104X
    [4X[28X5^2): x1^5*x2+x1*x2^5+Z(5)*x3^5*x4+Z(5)*x3*x4^5=0>>[128X[104X
    [4X[25Xgap>[125X [27Xhom := Intertwiner(duality);[127X[104X
    [4X[28XMappingByFunction( PDeltaO-(6,5), <projective collineation group of size [128X[104X
    [4X[28X58968000000 with 3 generators>, function( g ) ... end, function( g ) ... end )[128X[104X
    [4X[25Xgap>[125X [27Xg := Random(CollineationGroup(q5q));[127X[104X
    [4X[28X< a collineation: <cmat 6x6 over GF(5,1)>, F^0>[128X[104X
    [4X[25Xgap>[125X [27Xg^hom;[127X[104X
    [4X[28X< a collineation: <cmat 4x4 over GF(5,2)>, F^5>[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  combination of the isomorphism of the GQs [22XW(3,q)[122X, [22XQ(4,q)[122X when [22Xq[122X is even
  and  the  duality  between the same GQs, yields a duality from each of these
  GQs itself. The operation [11XSelfDuality[111X implements this combination.[133X
  
  [1X10.3-6 SelfDuality[101X
  
  [33X[1;0Y[29X[2XSelfDuality[102X( [3Xgq[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XSelfDuality[102X( [3Xgq[103X, [3Xboolean[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya geometry morphism[133X
  
  [33X[0;0YIt  is  checked  whether the base field of [3Xgq[103X is a field of characteristic 2
  and  whether  [3Xgq[103X  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 [3Xfalse[103X will return a geometry
  morphism that is not equipped with an intertwiner. The example shows the use
  of the boolean argument.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xq := 16;[127X[104X
    [4X[28X16[128X[104X
    [4X[25Xgap>[125X [27Xmat := [[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;[127X[104X
    [4X[28X[ [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ], [128X[104X
    [4X[28X  [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], [128X[104X
    [4X[28X  [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ], [128X[104X
    [4X[28X  [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], [128X[104X
    [4X[28X  [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ] ][128X[104X
    [4X[25Xgap>[125X [27Xform := QuadraticFormByMatrix(mat,GF(q));[127X[104X
    [4X[28X< quadratic form >[128X[104X
    [4X[25Xgap>[125X [27Xq4q := PolarSpace(form);[127X[104X
    [4X[28X<polar space in ProjectiveSpace(4,GF(2^4)): x_1*x_2+x_3^2+x_4*x_5=0 >[128X[104X
    [4X[25Xgap>[125X [27Xem := SelfDuality(q4q);[127X[104X
    [4X[28X#I  No intertwiner computed. The polar space must have a collineation group computed[128X[104X
    [4X[28X<geometry morphism from <Elements of Q(4, [128X[104X
    [4X[28X16): x_1*x_2+x_3^2+x_4*x_5=0> to <Elements of Q(4, [128X[104X
    [4X[28X16): x_1*x_2+x_3^2+x_4*x_5=0>>[128X[104X
    [4X[25Xgap>[125X [27XCollineationGroup(q4q);[127X[104X
    [4X[28X#I  Computing collineation group of canonical polar space...[128X[104X
    [4X[28X<projective collineation group of size 4380799795200 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27Xem := SelfDuality(q4q);[127X[104X
    [4X[28X<geometry morphism from <Elements of Q(4, [128X[104X
    [4X[28X16): x_1*x_2+x_3^2+x_4*x_5=0> to <Elements of Q(4, [128X[104X
    [4X[28X16): x_1*x_2+x_3^2+x_4*x_5=0>>[128X[104X
    [4X[25Xgap>[125X [27Xhom := Intertwiner(em);[127X[104X
    [4X[28XMappingByFunction( <projective collineation group of size 4380799795200 with [128X[104X
    [4X[28X3 generators>, <projective collineation group of size 4380799795200 with [128X[104X
    [4X[28X3 generators>, function( el ) ... end, function( el ) ... end )[128X[104X
    [4X[25Xgap>[125X [27Xq := 16;[127X[104X
    [4X[28X16[128X[104X
    [4X[25Xgap>[125X [27Xw := SymplecticSpace(3,q);[127X[104X
    [4X[28XW(3, 16)[128X[104X
    [4X[25Xgap>[125X [27Xem := SelfDuality(w);[127X[104X
    [4X[28X<geometry morphism from <Elements of W(3, 16)> to <Elements of W(3, 16)>>[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X10.4 [33X[0;0YEmbeddings of projective spaces[133X[101X
  
  [33X[0;0YThe  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.[133X
  
  [1X10.4-1 NaturalEmbeddingBySubspace[101X
  
  [33X[1;0Y[29X[2XNaturalEmbeddingBySubspace[102X( [3Xgeom1[103X, [3Xgeom2[103X, [3Xv[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya geometry morphism[133X
  
  [33X[0;0YThe  arguments  [3Xgeom1[103X  and  [3Xgeom2[103X  are  both  projective spaces, and [3Xv[103X is an
  element  of  a geom2. This function returns a geometry morphism representing
  the natural embedding of [3Xgeom1[103X into [3Xgeom2[103X as the subspace [3Xv[103X. Hence [3Xgeom1[103X and
  [3Xv[103X  must  be  equivalent as geometries. An Intertwiner is not implemented for
  this geometry morphism.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgeom1 := ProjectiveSpace(2, 3);[127X[104X
    [4X[28XProjectiveSpace(2, 3)[128X[104X
    [4X[25Xgap>[125X [27Xgeom2 := ProjectiveSpace(3, 3);[127X[104X
    [4X[28XProjectiveSpace(3, 3)[128X[104X
    [4X[25Xgap>[125X [27Xplanes := Planes(geom2);[127X[104X
    [4X[28X<planes of ProjectiveSpace(3, 3)>[128X[104X
    [4X[25Xgap>[125X [27Xhyp := Random(planes);[127X[104X
    [4X[28X<a plane in ProjectiveSpace(3, 3)>[128X[104X
    [4X[25Xgap>[125X [27Xem := NaturalEmbeddingBySubspace(geom1, geom2, hyp);[127X[104X
    [4X[28X<geometry morphism from <All elements of ProjectiveSpace(2, [128X[104X
    [4X[28X3)> to <All elements of ProjectiveSpace(3, 3)>>[128X[104X
    [4X[25Xgap>[125X [27Xpoints := Points(geom1);[127X[104X
    [4X[28X<points of ProjectiveSpace(2, 3)>[128X[104X
    [4X[25Xgap>[125X [27Xx := Random(points);[127X[104X
    [4X[28X<a point in ProjectiveSpace(2, 3)>[128X[104X
    [4X[25Xgap>[125X [27Xx^em;[127X[104X
    [4X[28X<a point in ProjectiveSpace(3, 3)>[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X10.4-2 NaturalEmbeddingBySubField[101X
  
  [33X[1;0Y[29X[2XNaturalEmbeddingBySubField[102X( [3Xgeom1[103X, [3Xgeom2[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XNaturalEmbeddingBySubField[102X( [3Xgeom1[103X, [3Xgeom2[103X, [3Xboolean[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya geometry morphism[133X
  
  [33X[0;0YThe  arguments  [3Xgeom1[103X and [3Xgeom2[103X are projective spaces of the same dimension.
  This function returns a geometry morphism representing the natural embedding
  of [3Xgeom1[103X into [3Xgeom2[103X as a subfield geometry. The geometry morphism also comes
  equipped  with an intertwiner (see [2XIntertwiner[102X ([14X10.1-2[114X)). The optional third
  argument  [3Xboolean[103X  can  take  either  [10Xtrue[110X  or  [10Xfalse[110X 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 [10Xtrue[110X. Note that
  the  source  of  the  intertwiner is the projectivity group of [3Xgeom1[103X and its
  range  is  a  subgroup  of the projectivity group of [3Xgeom2[103X. Here is a simple
  example where the geometry morphism embeds PG(2,3) into PG(2,9).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpg1 := PG(2,3);[127X[104X
    [4X[28XProjectiveSpace(2, 3)[128X[104X
    [4X[25Xgap>[125X [27Xpg2 := PG(2,9);[127X[104X
    [4X[28XProjectiveSpace(2, 9)[128X[104X
    [4X[25Xgap>[125X [27Xem := NaturalEmbeddingBySubfield(pg1,pg2);[127X[104X
    [4X[28X<geometry morphism from <All elements of ProjectiveSpace(2, [128X[104X
    [4X[28X3)> to <All elements of ProjectiveSpace(2, 9)>>[128X[104X
    [4X[25Xgap>[125X [27Xpoints := AsList(Points( pg1 ));[127X[104X
    [4X[28X[ <a point in ProjectiveSpace(2, 3)>, <a point in ProjectiveSpace(2, 3)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(2, 3)>, <a point in ProjectiveSpace(2, 3)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(2, 3)>, <a point in ProjectiveSpace(2, 3)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(2, 3)>, <a point in ProjectiveSpace(2, 3)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(2, 3)>, <a point in ProjectiveSpace(2, 3)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(2, 3)>, <a point in ProjectiveSpace(2, 3)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(2, 3)> ][128X[104X
    [4X[25Xgap>[125X [27Ximage := ImagesSet(em, points);[127X[104X
    [4X[28X[ <a point in ProjectiveSpace(2, 9)>, <a point in ProjectiveSpace(2, 9)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(2, 9)>, <a point in ProjectiveSpace(2, 9)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(2, 9)>, <a point in ProjectiveSpace(2, 9)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(2, 9)>, <a point in ProjectiveSpace(2, 9)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(2, 9)>, <a point in ProjectiveSpace(2, 9)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(2, 9)>, <a point in ProjectiveSpace(2, 9)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(2, 9)> ][128X[104X
    [4X[25Xgap>[125X [27Xhom := Intertwiner(em);[127X[104X
    [4X[28XMappingByFunction( The FinInG projectivity group PGL(3,3), <projective colline[128X[104X
    [4X[28Xation group of size 5616 with [128X[104X
    [4X[28X2 generators>, function( x ) ... end, function( y ) ... end )[128X[104X
    [4X[25Xgap>[125X [27Xgroup1 := ProjectivityGroup(pg1);[127X[104X
    [4X[28XThe FinInG projectivity group PGL(3,3)[128X[104X
    [4X[25Xgap>[125X [27Xgens := GeneratorsOfGroup(group1);[127X[104X
    [4X[28X[ < a collineation: <cmat 3x3 over GF(3,1)>, F^0>, [128X[104X
    [4X[28X  < a collineation: <cmat 3x3 over GF(3,1)>, F^0> ][128X[104X
    [4X[25Xgap>[125X [27Xgroup1_image := Group(List(gens,x->x^hom));[127X[104X
    [4X[28X<projective collineation group with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XOrder(group1_image);[127X[104X
    [4X[28X5616[128X[104X
    [4X[25Xgap>[125X [27Xgroup2 := ProjectivityGroup(pg2);[127X[104X
    [4X[28XThe FinInG projectivity group PGL(3,9)[128X[104X
    [4X[25Xgap>[125X [27XOrder(group2);[127X[104X
    [4X[28X42456960[128X[104X
    [4X[25Xgap>[125X [27Xg := Random(group2);[127X[104X
    [4X[28X< a collineation: <cmat 3x3 over GF(3,2)>, F^0>[128X[104X
    [4X[25Xgap>[125X [27XPreImageElm(hom,g);[127X[104X
    [4X[28X#I  <el> is not in the range of the intertwiner[128X[104X
    [4X[28Xfail[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X10.4-3 [33X[0;0YEmbedding of projective spaces by field reduction[133X[101X
  
  [33X[0;0YWe briefly describe the mathematics behind field reduction. For more details
  we  refer  to  [LVdV13].  Consider  the  fields [22XGF(q)[122X and [22XGF(q^t)[122X. The field
  [22XGF(q^t)[122X is a [22Xt[122X-dimensional vector space over [22XGF(q)[122X. Hence, with respect to a
  chosen  basis  [22XB[122X  for [22XGF(q^t)[122X as a [22XGF(q)[122X-vector space, the bijection between
  the  vector  spaces  [22XV(n,q^t)[122X  and  [22XV(tn,q)[122X can be implemented. Consider the
  projective  space  [22XPG(n-1,q^t)[122X. The elements are represented by subspaces of
  [22XV(n,q^t)[122X.   Clearly,  a  [22Xk[122X  dimensional  subspace  of  [22XV(n,q^t)[122X  is  also  a
  [22Xkn[122X-dimensional   subspace   of  [22XV(nt,q)[122X.  This  induces  an  embedding  from
  [22XPG(n-1,q^t)[122X  into [22XPG(nt-1,q)[122X. The embedding will be determined by the chosen
  basis of [22XGF(q^t)[122X as a vector space over [22XGF(q)[122X[133X
  
  [1X10.4-4 BlownUpSubspaceOfProjectiveSpace[101X
  
  [33X[1;0Y[29X[2XBlownUpSubspaceOfProjectiveSpace[102X( [3XB[103X, [3Xsubspace[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya subspace of a projective space[133X
  
  [33X[0;0YLet  [3XB[103X  be  a  basis  for  the  field [22XGF(q^t)[122X as [22XGF(q)[122X vector space, and let
  [3Xsubspace[103X  be  a  [22Xk-1[122X-dimensional  subspace  of  [22XPG(n-1,q^t)[122X represented by a
  [22Xk[122X-dimensional   subspace   [22XS[122X   of   [22XV(n,q^t)[122X.  This  operation  returns  the
  [22Xkt-1[122X-dimensional  subspace  of  [22XPG(nt-1,q)[122X  represented by blowing up [22XS[122X with
  respect  to  the  base  [3XB[103X.  This  operation  relies  on  the  [5XGAP[105X  operation
  [11XBlownUpMat[111X.  In  the  example,  the  effect of choosing a different basis is
  shown.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpg := PG(3,5^2);[127X[104X
    [4X[28XProjectiveSpace(3, 25)[128X[104X
    [4X[25Xgap>[125X [27Xbasis := Basis(AsVectorSpace(GF(5),GF(5^2)));[127X[104X
    [4X[28XCanonicalBasis( GF(5^2) )[128X[104X
    [4X[25Xgap>[125X [27Xline := Random(Lines(pg));[127X[104X
    [4X[28X<a line in ProjectiveSpace(3, 25)>[128X[104X
    [4X[25Xgap>[125X [27Xsolid1 := BlownUpSubspaceOfProjectiveSpace(basis,line);[127X[104X
    [4X[28X<a solid in ProjectiveSpace(7, 5)>[128X[104X
    [4X[25Xgap>[125X [27XBasisVectors(basis);[127X[104X
    [4X[28X[ Z(5)^0, Z(5^2) ][128X[104X
    [4X[25Xgap>[125X [27Xbasis := Basis(AsVectorSpace(GF(5),GF(5^2)),[Z(5),Z(5^2)^8]);[127X[104X
    [4X[28XBasis( GF(5^2), [ Z(5), Z(5^2)^8 ] )[128X[104X
    [4X[25Xgap>[125X [27Xsolid2 := BlownUpSubspaceOfProjectiveSpace(basis,line);[127X[104X
    [4X[28X<a solid in ProjectiveSpace(7, 5)>[128X[104X
    [4X[25Xgap>[125X [27Xsolid1 = solid2;[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X10.4-5 NaturalEmbeddingByFieldReduction[101X
  
  [33X[1;0Y[29X[2XNaturalEmbeddingByFieldReduction[102X( [3Xgeom1[103X, [3Xf2[103X, [3XB[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XNaturalEmbeddingByFieldReduction[102X( [3Xgeom1[103X, [3Xf2[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XNaturalEmbeddingByFieldReduction[102X( [3Xgeom1[103X, [3Xgeom2[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XNaturalEmbeddingByFieldReduction[102X( [3Xgeom1[103X, [3Xgeom2[103X, [3XB[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya geometry morphism[133X
  
  [33X[0;0YThis  operation  comes in four flavours. For the first flavour, the argument
  [3Xgeom1[103X  is  a  projective  space  over  a field [22XGF(q^t)[122X. The argument [3Xf2[103X is a
  subfield  [22XGF(q)[122X  of  [22XGF(q^t)[122X.  The  argument  [22XB[122X  is a basis for [22XGF(q^t)[122X as a
  [22XGF(q)[122X-vector  space.  When  this  argument is not given, a basis for [22XGF(q^t)[122X
  over  [22XGF(q)[122X  is  computed  using [11XBasis(Asvector space(GF(q),GF(q^t)))[111X. It is
  checked  whether  [3Xf2[103X is a subfield of the base field of [3Xgeom1[103X. The third and
  fourth flavour are comparable, where now [22XGF(q)[122X is found as the base field of
  [3Xgeom2[103X.  In  fact  the  arguments  [3Xgeom1[103X  and [3Xgeom2[103X are the projective spaces
  [22XPG(n-1,q^t)[122X  and  [22XPG(nt-1,q)[122X  respectively. As in the previous flavours, the
  argument [3XB[103X is optional.[133X
  
  [33X[0;0YAn  intertwiner  is  always  available  for  this geometry morphism, and has
  source  the  homography  group  of  [3Xgeom1[103X  and  as  range  a subgroup of the
  homography  group  of  [3Xgeom2[103X  (or  the  projective  space of the appropriate
  dimension  over [3Xf2[103X. 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.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpg1 := ProjectiveSpace(2,81);[127X[104X
    [4X[28XProjectiveSpace(2, 81)[128X[104X
    [4X[25Xgap>[125X [27Xf2 := GF(9);[127X[104X
    [4X[28XGF(3^2)[128X[104X
    [4X[25Xgap>[125X [27Xem := NaturalEmbeddingByFieldReduction(pg1,f2);[127X[104X
    [4X[28X<geometry morphism from <All elements of ProjectiveSpace(2, [128X[104X
    [4X[28X81)> to <All elements of ProjectiveSpace(5, 9)>>[128X[104X
    [4X[25Xgap>[125X [27Xf2 := GF(3);[127X[104X
    [4X[28XGF(3)[128X[104X
    [4X[25Xgap>[125X [27Xem := NaturalEmbeddingByFieldReduction(pg1,f2);[127X[104X
    [4X[28X<geometry morphism from <All elements of ProjectiveSpace(2, [128X[104X
    [4X[28X81)> to <All elements of ProjectiveSpace(11, 3)>>[128X[104X
    [4X[25Xgap>[125X [27Xpg2 := ProjectiveSpace(11,3);[127X[104X
    [4X[28XProjectiveSpace(11, 3)[128X[104X
    [4X[25Xgap>[125X [27Xem := NaturalEmbeddingByFieldReduction(pg1,pg2);[127X[104X
    [4X[28X<geometry morphism from <All elements of ProjectiveSpace(2, [128X[104X
    [4X[28X81)> to <All elements of ProjectiveSpace(11, 3)>>[128X[104X
    [4X[25Xgap>[125X [27Xpg1 := PG(1,9);[127X[104X
    [4X[28XProjectiveSpace(1, 9)[128X[104X
    [4X[25Xgap>[125X [27Xem := NaturalEmbeddingByFieldReduction(pg1,GF(3));[127X[104X
    [4X[28X<geometry morphism from <All elements of ProjectiveSpace(1, [128X[104X
    [4X[28X9)> to <All elements of ProjectiveSpace(3, 3)>>[128X[104X
    [4X[25Xgap>[125X [27Xi := Intertwiner(em);[127X[104X
    [4X[28XMappingByFunction( The FinInG projectivity group PGL(2,9), <projective colline[128X[104X
    [4X[28Xation group of size 720 with [128X[104X
    [4X[28X2 generators>, function( m ) ... end, function( m ) ... end )[128X[104X
    [4X[25Xgap>[125X [27Xspread := List(Points(pg1),x->x^em);[127X[104X
    [4X[28X[ <a line in ProjectiveSpace(3, 3)>, <a line in ProjectiveSpace(3, 3)>, [128X[104X
    [4X[28X  <a line in ProjectiveSpace(3, 3)>, <a line in ProjectiveSpace(3, 3)>, [128X[104X
    [4X[28X  <a line in ProjectiveSpace(3, 3)>, <a line in ProjectiveSpace(3, 3)>, [128X[104X
    [4X[28X  <a line in ProjectiveSpace(3, 3)>, <a line in ProjectiveSpace(3, 3)>, [128X[104X
    [4X[28X  <a line in ProjectiveSpace(3, 3)>, <a line in ProjectiveSpace(3, 3)> ][128X[104X
    [4X[25Xgap>[125X [27Xstab := Stabilizer(CollineationGroup(PG(3,3)),Set(spread),OnSets);[127X[104X
    [4X[28X<projective collineation group of size 5760 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27Xhom := HomographyGroup(pg1);[127X[104X
    [4X[28XThe FinInG projectivity group PGL(2,9)[128X[104X
    [4X[25Xgap>[125X [27Xgens := GeneratorsOfGroup(hom);;[127X[104X
    [4X[25Xgap>[125X [27Xgroup := Group(List(gens,x->x^i));[127X[104X
    [4X[28X<projective collineation group with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XOrder(group);[127X[104X
    [4X[28X2880[128X[104X
    [4X[25Xgap>[125X [27XIsSubgroup(stab,group);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  
  [1X10.5 [33X[0;0YEmbeddings of polar spaces[133X[101X
  
  [1X10.5-1 NaturalEmbeddingBySubspace[101X
  
  [33X[1;0Y[29X[2XNaturalEmbeddingBySubspace[102X( [3Xgeom1[103X, [3Xgeom2[103X, [3Xv[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya geometry morphism[133X
  
  [33X[0;0YThe  arguments  [3Xgeom1[103X  and [3Xgeom2[103X both polar spaces, and [3Xv[103X is an element of a
  projective space. This function returns a geometry morphism representing the
  natural  embedding  of [3Xgeom1[103X into the intersection of [3Xgeom2[103X and [3Xv[103X. Hence the
  intersection  of  [3Xgeom2[103X  and [3Xv[103X must induce a polar space of the same type as
  [3Xgeom1[103X  in [3Xv[103X. This operation performs all necessary checks. An Intertwiner is
  not implemented for this geometry morphism.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xh1 := HermitianPolarSpace(2, 3^2);[127X[104X
    [4X[28XH(2, 3^2)[128X[104X
    [4X[25Xgap>[125X [27Xh2 := HermitianPolarSpace(3, 3^2);[127X[104X
    [4X[28XH(3, 3^2)[128X[104X
    [4X[25Xgap>[125X [27Xpg := AmbientSpace( h2 );    [127X[104X
    [4X[28XProjectiveSpace(3, 9)[128X[104X
    [4X[25Xgap>[125X [27Xpi := VectorSpaceToElement( pg, [[1,0,0,0],[0,1,0,0],[0,0,1,0]] * Z(9)^0 );[127X[104X
    [4X[28X<a plane in ProjectiveSpace(3, 9)>[128X[104X
    [4X[25Xgap>[125X [27Xem := NaturalEmbeddingBySubspace( h1, h2, pi );[127X[104X
    [4X[28X<geometry morphism from <Elements of H(2, 3^2)> to <Elements of H(3, 3^2)>>[128X[104X
    [4X[25Xgap>[125X [27Xps1 := ParabolicQuadric(4,4);[127X[104X
    [4X[28XQ(4, 4)[128X[104X
    [4X[25Xgap>[125X [27Xps2 := ParabolicQuadric(6,4);[127X[104X
    [4X[28XQ(6, 4)[128X[104X
    [4X[25Xgap>[125X [27Xpg := AmbientSpace( ps2 );    [127X[104X
    [4X[28XProjectiveSpace(6, 4)[128X[104X
    [4X[25Xgap>[125X [27Xpi := VectorSpaceToElement( pg, [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[127X[104X
    [4X[25X>[125X [27X[0,0,0,1,0,0,0],[0,0,0,0,1,0,0]] * Z(4)^0 );[127X[104X
    [4X[28X<a proj. 4-space in ProjectiveSpace(6, 4)>[128X[104X
    [4X[25Xgap>[125X [27Xem := NaturalEmbeddingBySubspace( ps1, ps2, pi );[127X[104X
    [4X[28X<geometry morphism from <Elements of Q(4, 4)> to <Elements of Q(6, 4)>>[128X[104X
    [4X[25Xgap>[125X [27XList(Lines(ps1),x->x^em);[127X[104X
    [4X[28X[ <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>, [128X[104X
    [4X[28X  <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>, [128X[104X
    [4X[28X  <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>, [128X[104X
    [4X[28X  <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>, [128X[104X
    [4X[28X  <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>, [128X[104X
    [4X[28X  <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>, [128X[104X
    [4X[28X  <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>, [128X[104X
    [4X[28X  <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>, [128X[104X
    [4X[28X  <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>, [128X[104X
    [4X[28X  <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>, [128X[104X
    [4X[28X  <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>, [128X[104X
    [4X[28X  <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>, [128X[104X
    [4X[28X  <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>, [128X[104X
    [4X[28X  <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>, [128X[104X
    [4X[28X  <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>, [128X[104X
    [4X[28X  <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>, [128X[104X
    [4X[28X  <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>, [128X[104X
    [4X[28X  <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>, [128X[104X
    [4X[28X  <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>, [128X[104X
    [4X[28X  <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>, [128X[104X
    [4X[28X  <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>, [128X[104X
    [4X[28X  <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>, [128X[104X
    [4X[28X  <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>, [128X[104X
    [4X[28X  <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>, [128X[104X
    [4X[28X  <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>, [128X[104X
    [4X[28X  <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>, [128X[104X
    [4X[28X  <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>, [128X[104X
    [4X[28X  <a line in Q(6, 4)>, <a line in Q(6, 4)>, <a line in Q(6, 4)>, [128X[104X
    [4X[28X  <a line in Q(6, 4)> ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X10.5-2 NaturalEmbeddingBySubField[101X
  
  [33X[1;0Y[29X[2XNaturalEmbeddingBySubField[102X( [3Xgeom1[103X, [3Xgeom2[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XNaturalEmbeddingBySubField[102X( [3Xgeom1[103X, [3Xgeom2[103X, [3Xboolean[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya geometry morphism[133X
  
  [33X[0;0YThe  arguments  [3Xgeom1[103X  and  [3Xgeom2[103X  are  projective  or  polar spaces with an
  underlying  vector space of the same dimension and the base field [22XL[122X of [3Xgeom2[103X
  is  an  extension of the base field [22XK[122X of [3Xgeom1[103X. The form [22Xf[122X determining [3Xgeom1[103X
  also defines a form over [22XL[122X, and determines a polar space. By considering the
  underlying  vector  spaces  determining  the  elements  of  [3Xgeom1[103X  over  the
  extension field [22XL[122X, there is an obvious embedding of [3Xgeom1[103X in the polar space
  over  the extension field. Considering [22Xf[122X 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 [KL90]):[133X
  
        Polar Space 1 │ Polar Space 2 │ Conditions   
        ──────────────┼───────────────┼───────────  
        ──────────────┼───────────────┼───────────  
        [22XW(2n-1,q)[122X     │ [22XW(2n-1,q^a)[122X   │ --           
        [22XW(2n-1,q)[122X     │ [22XH(2n-1,q^a)[122X   │ --           
        [22XH(d,q^2)[122X      │ [22XH(d,q^(2r))[122X   │ [22Xr[122X odd        
        [22XO^ϵ(d,q)[122X      │ [22XH(d,q^2)[122X      │ [22Xq[122X odd        
        [22XO^ϵ(d,q)[122X      │ [22XO^ϵ'(d,q^r)[122X   │ [22Xϵ=(ϵ')^r[122X     
  
       [1XTable:[101X Subfield embeddings of polar spaces
  
  
  [33X[0;0YThe   geometry  morphism  also  comes  equipped  with  an  intertwiner  (see
  [2XIntertwiner[102X  ([14X10.1-2[114X)).  The optional third argument [3Xboolean[103X can take either
  [10Xtrue[110X  or [10Xfalse[110X as input, and then our operation will or will not compute the
  intertwiner  accordingly. When set [10Xtrue[110X, the intertwiner will be computed if
  [11XHasCollineationGroup(geom1)[111X  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 [10Xtrue[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xw := SymplecticSpace(5, 3);[127X[104X
    [4X[28XW(5, 3)[128X[104X
    [4X[25Xgap>[125X [27Xh := HermitianPolarSpace(5, 3^2);[127X[104X
    [4X[28XH(5, 3^2)[128X[104X
    [4X[25Xgap>[125X [27Xem := NaturalEmbeddingBySubfield(w, h);[127X[104X
    [4X[28X#I  No intertwiner computed. <geom1> must have a collineation group computed[128X[104X
    [4X[28X<geometry morphism from <Elements of W(5, 3)> to <Elements of H(5, 3^2)>>[128X[104X
    [4X[25Xgap>[125X [27Xpoints := AsList(Points(w));;[127X[104X
    [4X[25Xgap>[125X [27Ximage := ImagesSet(em, points);;[127X[104X
    [4X[25Xgap>[125X [27XForAll(image, x -> x in h);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xhq:=HyperbolicQuadric(3,4);[127X[104X
    [4X[28XQ+(3, 4)[128X[104X
    [4X[25Xgap>[125X [27Xeq:=EllipticQuadric(3,2);[127X[104X
    [4X[28XQ-(3, 2)[128X[104X
    [4X[25Xgap>[125X [27Xem:=NaturalEmbeddingBySubfield(eq,hq);[127X[104X
    [4X[28X#I  No intertwiner computed. <geom1> must have a collineation group computed[128X[104X
    [4X[28X<geometry morphism from <Elements of Q-(3, 2)> to <Elements of Q+(3, 4)>>[128X[104X
    [4X[25Xgap>[125X [27Xeqpts:=ImagesSet(em,AsList(Points(eq)));[127X[104X
    [4X[28X[ <a point in Q+(3, 4)>, <a point in Q+(3, 4)>, <a point in Q+(3, 4)>, [128X[104X
    [4X[28X  <a point in Q+(3, 4)>, <a point in Q+(3, 4)> ][128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  
  [1X10.5-3 [33X[0;0YEmbedding of polar spaces by field reduction[133X[101X
  
  [33X[0;0YField  reduction  for  polar  spaces  is  somewhat  more  involved  than for
  projective  spaces,  and  we  give  a  brief description. Let [22XL[122X be the field
  [22XGF(q^t)[122X  and let [22XK[122X be the field [22XGF(q)[122X Let [22XP[122X be a polar space over a field [22XL[122X.
  Let  [22Xf[122X be the form on the [22Xr[122X dimensional vector space [22XV[122X over [22XL[122X determining [22XP[122X.
  Consider the trace map[133X
  
  [33X[0;0Y[22XT: L -> K: x ↦ x^(q^t)+ x^(q^(t-1)) + ... + x.[122X[133X
  
  [33X[0;0YDefine for any [22Xα ∈ L[122X the map[133X
  
  [33X[0;0Y[22XT_α: L -> K: x ↦ T_α(x) = T(α x).[122X[133X
  
  [33X[0;0YConsider  the [22Xrt[122X dimensional vector space [22XW[122X over [22XK[122X. There is a bijective map
  [22XΦ[122X:  [22XV  ->  W[122X  and  [22XT_α  ∘  f ∘ Φ^-1[122X defines a quadratic or sesquilinear form
  (depending  on  [22Xα[122X,  and  [22Xf[122X being quadratic or sesquilinear) acting on [22XW[122X, and
  hence,  if  not  singular  or  degenerate, inducing a polar space [22XS[122X over the
  finite  field  [22XGF(q)[122X.  An element of [22XP[122X can be mapped onto an element of [22XS[122X by
  simply  blowing  up  [22XP[122X  using  field reduction for projective spaces. So the
  resulting polar space [22XS[122X is dependent on the original form [22Xf[122X, the parameter [22Xα[122X
  and  the  blowing  up  of  elements  by  field  reduction,  the latter being
  dependent  on  the  basis  of  [22XL[122X  as  a  [22XK[122X vector space. [5XFinInG[105X provides two
  approaches.  The  first approach starts from [22XP[122X and the parameters [22XK[122X, [22Xα[122X and a
  basis  for  [22XL[122X  as  [22XK[122X vector space. Then the resulting form [22XT_α ∘ f ∘ Φ^-1[122X is
  determined,  and  the  associated  polar  space  [22XS[122X  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 [22XP[122X and [22XS[122X.
  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 [22XP[122X to [22XS[122X. Note
  that  the  polar spaces used as an argument may be freely chosen and are not
  required to be in the canonical form.[133X
  
  [33X[0;0YFor  more  information on the embeddings by field reduction of polar spaces,
  including  conditions  on the parameter [22Xα[122X, we refer to [Gil08] and [LVdV13].
  The possible embeddings are listed in the following table.[133X
  
        Polar Space 1  │ Polar Space 2         │ Conditions    
        ───────────────┼───────────────────────┼────────────  
        ───────────────┼───────────────────────┼────────────  
        [22XW(2n-1,q^t)[122X    │ [22XW(2nt-1,q)[122X            │ --            
        [22XQ^+(2n-1,q^t)[122X  │ [22XQ^+(2nt-1,q)[122X          │ --            
        [22XQ^-(2n-1,q^t)[122X  │ [22XQ^-(2nt-1,q)[122X          │ --            
        [22XQ(2n,q^(2a+1))[122X │ [22XQ((2a+1)(2n+1)-1,q)[122X   │ [22Xq[122X odd         
        [22XQ(2n,q^(2a))[122X   │ [22XQ^-(2a(2n+1)-1,q)[122X     │ [22Xq ≡ 1 mod 4[122X   
        [22XQ(2n,q^(4a+2))[122X │ [22XQ^+((4a+2)(2n+1)-1,q)[122X │ [22Xq ≡ 3 mod 4[122X   
        [22XQ(2n,q^(4a))[122X   │ [22XQ^-(4a(2n+1)-1,q)[122X     │ [22Xq ≡ 3 mod 4[122X   
        [22XH(n,q^(2a+1))[122X  │ [22XH((2a+1)(n+1)-1,q)[122X    │ [22Xq[122X square      
        [22XH(n,q^(2a))[122X    │ [22XW(2a(n+1)-1,q)[122X        │ [22Xq[122X even        
        [22XH(2n,q^(2a))[122X   │ [22XQ^-(2a(2n+1)-1,q)[122X     │ [22Xq[122X odd         
        [22XH(2n+1,q^(2a))[122X │ [22XQ^+(2a(2n+2)-1,q)[122X     │ [22Xq[122X odd         
  
       [1XTable:[101X Field reduction of polar spaces
  
  
  [1X10.5-4 NaturalEmbeddingByFieldReduction[101X
  
  [33X[1;0Y[29X[2XNaturalEmbeddingByFieldReduction[102X( [3Xps1[103X, [3Xf2[103X, [3Xalpha[103X, [3Xbasis[103X, [3Xbool[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XNaturalEmbeddingByFieldReduction[102X( [3Xps1[103X, [3Xf2[103X, [3Xalpha[103X, [3Xbasis[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XNaturalEmbeddingByFieldReduction[102X( [3Xps1[103X, [3Xf2[103X, [3Xalpha[103X, [3Xbool[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XNaturalEmbeddingByFieldReduction[102X( [3Xps1[103X, [3Xf2[103X, [3Xalpha[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XNaturalEmbeddingByFieldReduction[102X( [3Xps1[103X, [3Xf2[103X, [3Xbool[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XNaturalEmbeddingByFieldReduction[102X( [3Xps1[103X, [3Xf2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya geometry morphism[133X
  
  [33X[0;0Y[3Xps1[103X  is a polar space over a field extension [22XL[122X of [3Xf2[103X, [3Xbasis[103X is a basis for [22XL[122X
  over   [3Xf2[103X,   [3Xalpha[103X   is   a   non-zero   element   of   [22XL[122X.  The  version  of
  [11XNaturalEmbeddingByFieldReduction[111X  implements the first approach as explained
  in  [14X10.5-3[114X.  When  no  argument  [3Xbasis[103X  is  given,  a basis for [22XL[122X over [3Xf2[103X is
  computed  using [11XBasis(Asvector space(K,L))[111X. When no argument [3Xalpha[103X is given,
  [11XOne(f2)[111X  is  used  as  value  for  [3Xalpha[103X. When [3Xbool[103X is true or not given, an
  intertwiner  is  computed,  when  [3Xbool[103X is [3Xfalse[103X, no intertwiner is computed.
  This  intertwiner  has as its domain the [12Xisometry group[112X of [3Xps1[103X. 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 [10Xtrue[110X.
  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).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xps1 := SymplecticSpace(1,3^3);[127X[104X
    [4X[28XW(1, 27)[128X[104X
    [4X[25Xgap>[125X [27Xem := NaturalEmbeddingByFieldReduction(ps1,GF(3),true);[127X[104X
    [4X[28X<geometry morphism from <Elements of W(1, [128X[104X
    [4X[28X27)> to <Elements of <polar space in ProjectiveSpace([128X[104X
    [4X[28X5,GF(3)): -x1*y6-x2*y5-x3*y4-x3*y6+x4*y3+x5*y2+x6*y1+x6*y3=0 >>>[128X[104X
    [4X[25Xgap>[125X [27Xps2 := AmbientGeometry(Range(em));[127X[104X
    [4X[28X<polar space in ProjectiveSpace([128X[104X
    [4X[28X5,GF(3)): -x1*y6-x2*y5-x3*y4-x3*y6+x4*y3+x5*y2+x6*y1+x6*y3=0 >[128X[104X
    [4X[25Xgap>[125X [27Xspread := List(Points(ps1),x->x^em);;[127X[104X
    [4X[25Xgap>[125X [27Xi := Intertwiner(em);[127X[104X
    [4X[28XMappingByFunction( PSp(2,27), <projective collineation group of size [128X[104X
    [4X[28X9828 with 2 generators>, function( m ) ... end, function( m ) ... end )[128X[104X
    [4X[25Xgap>[125X [27Xcoll := CollineationGroup(ps2);[127X[104X
    [4X[28X#I  Computing collineation group of canonical polar space...[128X[104X
    [4X[28X<projective collineation group of size 9170703360 with 4 generators>[128X[104X
    [4X[25Xgap>[125X [27Xstab := Group(ImagesSet(i,GeneratorsOfGroup(IsometryGroup(ps1))));[127X[104X
    [4X[28X<projective collineation group with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsSubgroup(coll,stab);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XList(Orbit(stab,spread[1]),x->x in spread);[127X[104X
    [4X[28X[ true, true, true, true, true, true, true, true, true, true, true, true, [128X[104X
    [4X[28X  true, true, true, true, true, true, true, true, true, true, true, true, [128X[104X
    [4X[28X  true, true, true, true ][128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xps1 := SymplecticSpace(3,9);[127X[104X
    [4X[28XW(3, 9)[128X[104X
    [4X[25Xgap>[125X [27Xem := NaturalEmbeddingByFieldReduction(ps1,GF(3),true);[127X[104X
    [4X[28X<geometry morphism from <Elements of W(3, [128X[104X
    [4X[28X9)> to <Elements of <polar space in ProjectiveSpace([128X[104X
    [4X[28X7[128X[104X
    [4X[28X ,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[128X[104X
    [4X[28X*y5=0 >>>[128X[104X
    [4X[25Xgap>[125X [27Xps2 := AmbientGeometry(Range(em));[127X[104X
    [4X[28X<polar space in ProjectiveSpace([128X[104X
    [4X[28X7[128X[104X
    [4X[28X ,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[128X[104X
    [4X[28X*y5=0 >[128X[104X
    [4X[25Xgap>[125X [27Xpg := AmbientSpace(ps2);[127X[104X
    [4X[28XProjectiveSpace(7, 3)[128X[104X
    [4X[25Xgap>[125X [27Xspread := List(Points(ps1),x->x^em);;[127X[104X
    [4X[25Xgap>[125X [27Xel := Random(ElementsOfIncidenceStructure(pg,5));[127X[104X
    [4X[28X<a proj. 4-space in ProjectiveSpace(7, 3)>[128X[104X
    [4X[25Xgap>[125X [27Xprebs := Filtered(spread,x->Meet(x,el) <> EmptySubspace(pg));;[127X[104X
    [4X[25Xgap>[125X [27Xbs := List(prebs,x->PreImageElm(em,x));;[127X[104X
    [4X[25Xgap>[125X [27XLength(bs);[127X[104X
    [4X[28X118[128X[104X
    [4X[25Xgap>[125X [27Xlines := List(Lines(ps1));;[127X[104X
    [4X[25Xgap>[125X [27XCollected(List(lines,x->Length(Filtered(bs,y->y * x))));[127X[104X
    [4X[28X[ [ 1, 702 ], [ 4, 117 ], [ 10, 1 ] ][128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X10.5-5 NaturalEmbeddingByFieldReduction[101X
  
  [33X[1;0Y[29X[2XNaturalEmbeddingByFieldReduction[102X( [3Xps1[103X, [3Xps2[103X, [3Xbool[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XNaturalEmbeddingByFieldReduction[102X( [3Xps1[103X, [3Xps2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya geometry morphism[133X
  
  [33X[0;0YIf  [3Xps1[103X  and [3Xps2[103X are two polar spaces which are suitable for field reduction
  as listed in the table with possible embeddings in Section [14X10.5-3[114X, then this
  operation returns the corresponding embedding. An intertwiner is computed if
  the  third  argument  [3Xbool[103X  is  true, or if there is no third argument. This
  intertwiner  has  as its domain the [12Xisometry group[112X of [3Xps1[103X. The example shows
  two cases where a spread is computed, including a subgroup of its stabiliser
  group using the intertwiner.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xps1 := SymplecticSpace(1,5^3);[127X[104X
    [4X[28XW(1, 125)[128X[104X
    [4X[25Xgap>[125X [27Xps2 := SymplecticSpace(5,5);[127X[104X
    [4X[28XW(5, 5)[128X[104X
    [4X[25Xgap>[125X [27Xem := NaturalEmbeddingByFieldReduction(ps1,ps2);[127X[104X
    [4X[28X#I  These polar spaces are suitable for field reduction[128X[104X
    [4X[28X<geometry morphism from <Elements of W(1, 125)> to <Elements of W(5, 5)>>[128X[104X
    [4X[25Xgap>[125X [27Xpts := Points(ps1);[127X[104X
    [4X[28X<points of W(1, 125)>[128X[104X
    [4X[25Xgap>[125X [27Xspread := List(pts,x->x^em);;[127X[104X
    [4X[25Xgap>[125X [27Xtest := Union(List(spread,x->List(Points(x))));;[127X[104X
    [4X[25Xgap>[125X [27XSet(test)=Set(AsList(Points(ps2)));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xhom := Intertwiner(em);[127X[104X
    [4X[28XMappingByFunction( PSp(2,125), <projective collineation group of size [128X[104X
    [4X[28X976500 with 2 generators>, function( m ) ... end, function( m ) ... end )[128X[104X
    [4X[25Xgap>[125X [27Xgroup := IsometryGroup(ps1);[127X[104X
    [4X[28XPSp(2,125)[128X[104X
    [4X[25Xgap>[125X [27XOrder(group);[127X[104X
    [4X[28X976500[128X[104X
    [4X[25Xgap>[125X [27Xgens := List(GeneratorsOfGroup(group),x->x^hom);[127X[104X
    [4X[28X[ < a collineation: <cmat 6x6 over GF(5,1)>, F^0>, [128X[104X
    [4X[28X  < a collineation: <cmat 6x6 over GF(5,1)>, F^0> ][128X[104X
    [4X[25Xgap>[125X [27Xgroup2 := Range(hom);[127X[104X
    [4X[28X<projective collineation group of size 976500 with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XOrder(group2);[127X[104X
    [4X[28X976500[128X[104X
    [4X[25Xgap>[125X [27Xbiggroup := CollineationGroup(ps2);[127X[104X
    [4X[28XPGammaSp(6,5)[128X[104X
    [4X[25Xgap>[125X [27Xstab := FiningSetwiseStabiliser(biggroup,spread);[127X[104X
    [4X[28X#I  Computing adjusted stabilizer chain...[128X[104X
    [4X[28X<projective collineation group with 7 generators>[128X[104X
    [4X[25Xgap>[125X [27Xtime;[127X[104X
    [4X[28X6907[128X[104X
    [4X[25Xgap>[125X [27XOrder(stab);[127X[104X
    [4X[28X5859000[128X[104X
    [4X[25Xgap>[125X [27Xps1 := HermitianPolarSpace(2,7^2);[127X[104X
    [4X[28XH(2, 7^2)[128X[104X
    [4X[25Xgap>[125X [27Xps2 := EllipticQuadric(5,7);[127X[104X
    [4X[28XQ-(5, 7)[128X[104X
    [4X[25Xgap>[125X [27Xem := NaturalEmbeddingByFieldReduction(ps1,ps2);[127X[104X
    [4X[28X#I  These polar spaces are suitable for field reduction[128X[104X
    [4X[28X<geometry morphism from <Elements of H(2, 7^2)> to <Elements of Q-(5, 7)>>[128X[104X
    [4X[25Xgap>[125X [27Xpts := Points(ps1);[127X[104X
    [4X[28X<points of H(2, 7^2)>[128X[104X
    [4X[25Xgap>[125X [27Xspread := List(pts,x->x^em);;[127X[104X
    [4X[25Xgap>[125X [27Xtest := Union(List(spread,x->List(Points(x))));;[127X[104X
    [4X[25Xgap>[125X [27XSet(test)=Set(AsList(Points(ps2)));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xhom := Intertwiner(em);[127X[104X
    [4X[28XMappingByFunction( PGU(3,7^2), <projective collineation group of size [128X[104X
    [4X[28X5663616 with 2 generators>, function( m ) ... end, function( m ) ... end )[128X[104X
    [4X[25Xgap>[125X [27Xgroup := IsometryGroup(ps1);[127X[104X
    [4X[28XPGU(3,7^2)[128X[104X
    [4X[25Xgap>[125X [27XOrder(group);[127X[104X
    [4X[28X5663616[128X[104X
    [4X[25Xgap>[125X [27Xgens := List(GeneratorsOfGroup(group),x->x^hom);[127X[104X
    [4X[28X[ < a collineation: <cmat 6x6 over GF(7,1)>, F^0>, [128X[104X
    [4X[28X  < a collineation: <cmat 6x6 over GF(7,1)>, F^0> ][128X[104X
    [4X[25Xgap>[125X [27Xgroup2 := Range(hom);[127X[104X
    [4X[28X<projective collineation group of size 5663616 with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XOrder(group2);[127X[104X
    [4X[28X5663616[128X[104X
    [4X[25Xgap>[125X [27Xbiggroup := CollineationGroup(ps2);[127X[104X
    [4X[28XPDeltaO-(6,7)[128X[104X
    [4X[25Xgap>[125X [27Xstab := FiningSetwiseStabiliser(biggroup,spread);[127X[104X
    [4X[28X#I  Computing adjusted stabilizer chain...[128X[104X
    [4X[28X<projective collineation group with 10 generators>[128X[104X
    [4X[25Xgap>[125X [27Xtime;[127X[104X
    [4X[28X3438[128X[104X
    [4X[25Xgap>[125X [27XOrder(stab);[127X[104X
    [4X[28X90617856[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X10.6 [33X[0;0YProjections[133X[101X
  
  [1X10.6-1 NaturalProjectionBySubspace[101X
  
  [33X[1;0Y[29X[2XNaturalProjectionBySubspace[102X( [3Xps[103X, [3Xv[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya geometry morphism[133X
  
  [33X[0;0YThe  argument  [3Xps[103X is a projective or polar space, and [3Xv[103X is a subspace of [3Xps[103X.
  In  the  case  that  [3Xps[103X  is  a  projective  space, the geometry of subspaces
  containing  [3Xv[103X  is  a  projective space of lower dimension over the same base
  field,  and  this  operation returns the corresponding geometry morphism. In
  the case that [3Xps[103X is a polar space, the geometry of elements pf [3Xps[103X containing
  [3Xv[103X  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 [3Xv[103X is a subspace of [3Xps[103X, 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,   [10Xps  /  v[110X  achieves  the  same  thing  as
  [10XAmbientGeometry(Range(NaturalProjectionBySubspace(ps,  v)))[110X.  An intertwiner
  is not available for this geometry morphism.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xps := HyperbolicQuadric(5,3);[127X[104X
    [4X[28XQ+(5, 3)[128X[104X
    [4X[25Xgap>[125X [27Xx := Random(Points(ps));;[127X[104X
    [4X[25Xgap>[125X [27Xplanes_on_x := AsList( Planes(x) );[127X[104X
    [4X[28X[ <a plane in Q+(5, 3)>, <a plane in Q+(5, 3)>, <a plane in Q+(5, 3)>, [128X[104X
    [4X[28X  <a plane in Q+(5, 3)>, <a plane in Q+(5, 3)>, <a plane in Q+(5, 3)>, [128X[104X
    [4X[28X  <a plane in Q+(5, 3)>, <a plane in Q+(5, 3)> ][128X[104X
    [4X[25Xgap>[125X [27Xproj := NaturalProjectionBySubspace(ps, x);[127X[104X
    [4X[28X<geometry morphism from <Elements of Q+(5, [128X[104X
    [4X[28X3)> to <Elements of <polar space in ProjectiveSpace([128X[104X
    [4X[28X3,GF(3)): x_1*x_2+x_3*x_4=0 >>>[128X[104X
    [4X[25Xgap>[125X [27Ximage := ImagesSet(proj, planes_on_x);[127X[104X
    [4X[28X[ <a line in Q+(3, 3): x_1*x_2+x_3*x_4=0>, [128X[104X
    [4X[28X  <a line in Q+(3, 3): x_1*x_2+x_3*x_4=0>, [128X[104X
    [4X[28X  <a line in Q+(3, 3): x_1*x_2+x_3*x_4=0>, [128X[104X
    [4X[28X  <a line in Q+(3, 3): x_1*x_2+x_3*x_4=0>, [128X[104X
    [4X[28X  <a line in Q+(3, 3): x_1*x_2+x_3*x_4=0>, [128X[104X
    [4X[28X  <a line in Q+(3, 3): x_1*x_2+x_3*x_4=0>, [128X[104X
    [4X[28X  <a line in Q+(3, 3): x_1*x_2+x_3*x_4=0>, [128X[104X
    [4X[28X  <a line in Q+(3, 3): x_1*x_2+x_3*x_4=0> ][128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  
  [1X10.7 [33X[0;0YProjective completion[133X[101X
  
  [1X10.7-1 ProjectiveCompletion[101X
  
  [33X[1;0Y[29X[2XProjectiveCompletion[102X( [3Xas[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya geometry morphism[133X
  
  [33X[0;0YThe  argument  [3Xas[103X is an affine space. This operation returns an embedding of
  [3Xas[103X  into  the  projective  space [3Xps[103X of the same dimension, and over the same
  field.  For example, the point [22X(x, y, z)[122X is mapped onto the projective point
  with  homogeneous  coordinates  [22X(1, x, y, z)[122X. An intertwiner is unnecessary,
  [11XCollineationGroup(as)[111X is a subgroup of [11XCollineationGroup(ps)[111X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xas := AffineSpace(3,5);[127X[104X
    [4X[28XAG(3, 5)[128X[104X
    [4X[25Xgap>[125X [27Xmap := ProjectiveCompletion(as);[127X[104X
    [4X[28X<geometry morphism from <Elements of AG(3, [128X[104X
    [4X[28X5)> to <All elements of ProjectiveSpace(3, 5)>>[128X[104X
    [4X[25Xgap>[125X [27Xp := Random( Points(as) );[127X[104X
    [4X[28X<a point in AG(3, 5)>[128X[104X
    [4X[25Xgap>[125X [27Xp^map;[127X[104X
    [4X[28X<a point in ProjectiveSpace(3, 5)>[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
