Impressum morphisms.gi Interaktion und
PortierbarkeitShell

#############################################################################
##
## morphisms.gi FinInG package
## John Bamberg
## Anton Betten
## Jan De Beule
## Philippe Cara
## Michel Lavrauw
## Max Neunhoeffer
##
## Copyright 2018 Colorado State University
## Sabancı Üniversitesi
## Università degli Studi di Padova
## Universiteit Gent
## University of St. Andrews
## University of Western Australia
## Vrije Universiteit Brussel
##
##
## Implementation stuff for incidence geometry morphisms
##
#############################################################################

########################################
# 20/3/14
# cmat notice: in many operations we will compute
# a matrix to initialize a form. Forms can not handle cmats, so
# Unpacking the cmats will sometimes be necessary.
########################################

############################################################
## Generic constructor operations, not intended for the user.
############################################################

## The three methods for GeometryMorphismByFunction are analogous
## with the MappingByFunction function. It behaves in exactly the same
## way except that we return an IsGeometryMorphism.

# CHECKED 27/09/11 jdb
#############################################################################
#O GeometryMorphismByFunction( <els1>, <els2>, <fun>, <bool>, <prefun> )
##
InstallMethod( GeometryMorphismByFunction,
"for two times a collection of elements of any type, a function, a boolean, and a function",
[ IsAnyElementsOfIncidenceStructure, IsAnyElementsOfIncidenceStructure,
IsFunction, IsBool, IsFunction ],
function( els1, els2, fun, bool, prefun )
 local morphism;
 morphism := MappingByFunction(els1, els2, fun, bool, prefun );
 SetFilterObj( morphism, IsGeometryMorphism );
 return morphism;
end );

# CHECKED 27/09/11 jdb
#############################################################################
#O GeometryMorphismByFunction( <els1>, <els2>, <fun>, <prefun> )
##
InstallMethod( GeometryMorphismByFunction,
"for two times a collection of elements of any type, and two times a function",
[ IsAnyElementsOfIncidenceStructure, IsAnyElementsOfIncidenceStructure,
 IsFunction, IsFunction ],
function( els1, els2, fun, inv )
 local morphism;
 morphism := MappingByFunction(els1, els2, fun, inv );
 SetFilterObj( morphism, IsGeometryMorphism );
 return morphism;
end );

# CHECKED 27/09/11 jdb
#############################################################################
#O GeometryMorphismByFunction( <els1>, <els2>, <fun> )
##
InstallMethod( GeometryMorphismByFunction,
"for two times a collection of elements of any type, and a function",
[ IsAnyElementsOfIncidenceStructure, IsAnyElementsOfIncidenceStructure,
 IsFunction ],
function( els1, els2, fun )
 local morphism;
 morphism := MappingByFunction(els1, els2, fun);
 SetFilterObj( morphism, IsGeometryMorphism );
 return morphism;
end );

#############################################################################
# Display methods
#############################################################################

InstallMethod( ViewObj,
"for a geometry morphism",
[ IsGeometryMorphism ],
function( f )
 Print("<geometry morphism from ");
 ViewObj(Source(f));
 Print( " to " );
 ViewObj(Range(f));
 Print(">");
end );

InstallMethod( PrintObj,
"for a geometry morphism",
[ IsGeometryMorphism ],
function( f )
 Print("Geometry morphism:\n ", f, "\n");
end );

InstallMethod( Display,
"for a geometry morphism",
[ IsGeometryMorphism ],
function( f )
 Print("Geometry morphism: ", Source(f), " -> ", Range(f), "\n");
end );

InstallMethod( ViewObj,
"for a geometry morphism",
[ IsGeometryMorphism and IsMappingByFunctionWithInverseRep ],
function( f )
 Print("<geometry morphism from ");
 ViewObj(Source(f));
 Print( " to " );
 ViewObj(Range(f));
 Print(">");
end );

InstallMethod( ViewObj,
"for a geometry morphism",
[ IsGeometryMorphism and IsMappingByFunctionRep ],
function( f )
 Print("<geometry morphism from ");
 ViewObj(Source(f));
 Print( " to " );
 ViewObj(Range(f));
 Print(">");
end );

InstallMethod( PrintObj,
"for a geometry morphism",
[ IsGeometryMorphism and IsMappingByFunctionRep ],
function( f )
 Print("Geometry morphism:\n ", f, "\n");
end );

InstallMethod( Display,
"for a geometry morphism",
[ IsGeometryMorphism and IsMappingByFunctionRep ],
function( f )
 Print("Geometry morphism: ", Source(f), " -> ", Range(f), "\n");
end );

############################################################
## Generic methods for the Image* operations for IsGeometryMorphism.
############################################################

# CHECKED 27/09/11 jdb
#############################################################################
#O ImageElm( , <x> )
##
InstallOtherMethod( ImageElm,
"for a geometry morphism and an element of an incidence structure",
[IsGeometryMorphism, IsElementOfIncidenceStructure],
function(em, x)
 return em!.fun(x);
end );

# CHECKED 27/09/11 jdb
#############################################################################
#O \^( <x>, )
##
InstallOtherMethod( \^,
"for an element of an incidence structure and a geometry morphism",
[IsElementOfIncidenceStructure, IsGeometryMorphism],
function(x, em)
 return ImageElm(em,x);
end );

# CHECKED 27/09/11 jdb
#############################################################################
#O ImagesSet( , <x> )
##
InstallOtherMethod( ImagesSet,
"for a geometry morphism and a collection of elements of an incidence structure",
[IsGeometryMorphism, IsElementOfIncidenceStructureCollection],
function(em, x)
 return List(x, t -> em!.fun(t));
end );

# CHECKED 27/09/11 jdb
#############################################################################
#O PreImageElm( , <x> )
##
InstallOtherMethod( PreImageElm,
"for a geometry morphism and an element of an incidence structure",
[IsGeometryMorphism, IsElementOfIncidenceStructure],
function(em, x)
 if IsInjective(em) then
 return em!.prefun(x);
 else
 Error("Map is not injective");
 fi;
end );

# CHECKED 27/09/11 jdb
#############################################################################
#O PreImagesSet( , <x> )
##
InstallOtherMethod( PreImagesSet,
"for a geometry morphism and an element of an incidence structure",
[IsGeometryMorphism, IsElementOfIncidenceStructureCollection],
function(em, x)
 return List(x, t -> em!.prefun(t));
end );

##########################################################
## User methods for the "natural geometry morphisms"
##########################################################

## The specialised operations...

# CHECKED 14/12/11 jdb + ml (and added a check that basefields of <ps1> and <ps2> are equal.)
# cmat changed 21/3/14.
#############################################################################
#O NaturalEmbeddingBySubspace( <ps1>, <ps2>, <v> ) returns a geometry morphism
# from the projective space <ps1> into <v>, an element of the projective space
# <ps2>. <v> must be of the right type, i.e. same projective dimension as <ps1>
# and <ps1> and <ps2> must be projective spaces over the same field.
##
InstallMethod( NaturalEmbeddingBySubspace,
"for a projective space into another, via a specified subspace",
[ IsProjectiveSpace, IsProjectiveSpace, IsSubspaceOfProjectiveSpace ],
function( geom1, geom2, v )
 local d1, d2, rk, f, invbasis, basis, func, pre, map, morphism, bs;
 rk := v!.type;
 basis := Unpack(v!.obj); #cmat change
 d1 := geom1!.dimension + 1;
 d2 := geom2!.dimension + 1;
 f := geom2!.basefield;
 if not v in geom2 then
 Error("Subspace is not an element of ", geom2);
 fi;
 if d2 < d1 or d1 <> rk then
 Error("Dimensions are incompatible");
 fi;
 if not f = geom1!.basefield then
 Error("Basefields must be the same");
 fi;
 ## To find the preimage, we first find an invertible matrix C such that
 ## [a1,..,ad]B = [a1,...,ad,0,...,0]C (where B is our d x e matrix "basis")
 ## is our embedding. We simply extend B to a basis for geom2.
 bs := BaseSteinitzVectors(Basis(geom2!.vectorspace), basis);
 invbasis := Inverse(Concatenation(bs!.subspace, bs!.factorspace));
 #ConvertToMatrixRep(invbasis, f); #useless now.
 func := x -> VectorSpaceToElement(geom2 , Unpack(x!.obj) * basis); #VectorSpaceToElement makes sure the 2nd arg becomes cmat.
 pre := function(y)
 local newy;
 if not y in v then
 Error("Applying preimage to an element which is not in the range");
 fi;
 newy:= Unpack(y!.obj) * invbasis; #cmat change.
 if not IsMatrix(newy) then
 newy := newy{[1..d1]};
 #ConvertToVectorRepNC(newy, f); #useless now.
 else
 newy := newy{[1..Size(newy)]}{[1..d1]};
 #ConvertToMatrixRepNC(newy, f); #useless now.
 fi;
 return VectorSpaceToElement(geom1, newy); #VectorSpaceToElement etc.
 end;
 morphism := GeometryMorphismByFunction(ElementsOfIncidenceStructure(geom1),
 ElementsOfIncidenceStructure(geom2),
 func, false, pre );
 SetIsInjective( morphism, true );
 return morphism;
end );

# CHECKED 27/09/11 jdb + ml
# cmat changed 21/3/14.
#############################################################################
#O NaturalEmbeddingBySubspaceNC( <ps1>, <ps2>, <v> )
## This operation is just like its namesake except that it
## has no checks
##
InstallMethod( NaturalEmbeddingBySubspaceNC,
"for a projective space into another, via a specified subspace",
[ IsProjectiveSpace, IsProjectiveSpace, IsSubspaceOfProjectiveSpace ],
function( geom1, geom2, v )
 local d1, f, invbasis, basis, func, pre, map, morphism, bs;
 basis := Unpack(v!.obj); #cmat change
 d1 := geom1!.dimension + 1;
 f := geom2!.basefield;
 bs := BaseSteinitzVectors(Basis(geom2!.vectorspace), basis);
 invbasis := Inverse(Concatenation(bs!.subspace, bs!.factorspace));
 # ConvertToMatrixRep(invbasis, f); #useless now.
 func := x -> VectorSpaceToElement(geom2 , x!.obj * basis); #VectorSpaceToElement etc.
 pre := function(y)
 local newy;
 newy:= Unpack(y!.obj) * invbasis; #cmat change
 if not IsMatrix(newy) then
 newy := newy{[1..d1]};
 #ConvertToVectorRepNC(newy, f); useless now.
 else
 newy := newy{[1..Size(newy)]}{[1..d1]};
 #ConvertToMatrixRepNC(newy, f); useless now.
 fi;
 return VectorSpaceToElement(geom1, newy); #cfr. supra.
 end;
 morphism := GeometryMorphismByFunction(ElementsOfIncidenceStructure(geom1),
 ElementsOfIncidenceStructure(geom2),
 func, false, pre );
 SetIsInjective( morphism, true );
 return morphism;
end );

# CHECKED 27/09/11 jdb + ml
# cmat version 20/3/14.
#############################################################################
#O NaturalEmbeddingBySubspace( <ps1>, <ps2>, <v> ) returns a geometry morphism
# from the polar space <ps1> into <ps2>, a polar space induced as a section of
# <ps1> and <v>, the latter a subspace of the ambient projective space of <ps1>
##
InstallMethod( NaturalEmbeddingBySubspace,
"for a polar space into another, via a specified subspace",
[ IsClassicalPolarSpace, IsClassicalPolarSpace, IsSubspaceOfProjectiveSpace ],
function( geom1, geom2, v )
local map, d1, d2, rk, f, i, basis, invbasis, bs, c1, c2, orth, tyv, quad1, quad2,
 change, perp2, formonv, ses1, ses2, ty1, ty2, newmat, func, pre, invchange;
 rk := v!.type;
 ty1 := PolarSpaceType(geom1);
 ty2 := PolarSpaceType(geom2);
 f := geom1!.basefield;
 d1 := geom1!.dimension;
 d2 := geom2!.dimension;
 tyv := TypeOfSubspace( geom2, v );
 ## Check that fields are the same and v is non-degenerate
 if geom2!.basefield <> f then
 Error("fields of both spaces must be the same");
 fi;
 #27/9/2011. jdb was wondering on the next 7 lines. Is this not just tyv = "degenerate"?
#if not (ty2 = "parabolic" and IsEvenInt(Size(f))) then
 # perp2 := Polarity(geom2);
 # if perp2(v) in geom2 then
 # Error("subspace is degenerate");
 # return;
 # fi;
 #fi;
 if rk > d2 or d1 > d2 then
 Error("dimensions are incompatible");
 fi;
 if tyv = "degenerate" then
 Error("subspace is degenerate");
 elif
 tyv <> ty1 then
 Error("non-degenerate section is not the same type as ", geom1);
 fi;
 orth := ["parabolic", "elliptic", "hyperbolic"];
 if ty1 = ty2 or (ty1 in orth and ty2 in orth) then

 ## Let B be basis of v. Then the form BM(B^T)^sigma (n.b. sigma
 ## is a field automorphism), where M is the form for geom2,
 ## will be equivalent to the form for geom1.

 basis := Unpack(v!.obj); #cmat unpack

 ## As usual we must consider two cases, the quadratic form case
 ## and the sesquilinear form case. We then obtain a base change matrix c1.

 if HasQuadraticForm( geom2 ) and HasQuadraticForm( geom1 ) then
 quad1 := QuadraticForm(geom1); #cmat notice: these are never cmats. So keep untouched.
 quad2 := QuadraticForm(geom2);
 newmat := basis * quad2!.matrix * TransposedMat(basis);
 formonv := QuadraticFormByMatrix(newmat, f);
 c1 := BaseChangeToCanonical( quad1 );
 else
 ses1 := SesquilinearForm(geom1);
 ses2 := SesquilinearForm(geom2);
 if ty1 = "hermitian" then
 newmat := basis * ses2!.matrix * (TransposedMat(basis))^CompanionAutomorphism(geom1);
 formonv := HermitianFormByMatrix(newmat, f);
 else
 newmat := basis * ses2!.matrix * TransposedMat(basis);
 formonv := BilinearFormByMatrix(newmat, f);
 fi;
 c1 := BaseChangeToCanonical( ses1 );
 fi;

 ## Finding isometry from geom1 to polar space defined by formofv:

 c2 := BaseChangeToCanonical( formonv );
 change := c1^-1 * c2;
 # ConvertToMatrixRep(change, f); #became useless.
 invchange := change^-1;
 bs := BaseSteinitzVectors(Basis(geom2!.vectorspace), basis);
 invbasis := Inverse(Concatenation(bs!.subspace, bs!.factorspace));
 # ConvertToMatrixRep(invbasis, f); #same here.
 func := x -> VectorSpaceToElement( geom2, Unpack(x!.obj) * change * basis);
 pre := function(y)
 local newy;
 if not y in v then
 Error("Applying preimage to an element which is not in the range");
 fi;
 newy:= Unpack(y!.obj) * invbasis;
 if IsMatrix(newy) then
 newy := newy{[1..Size(newy)]}{[1..rk]};
 ConvertToMatrixRepNC(newy, f);
 else
 newy := newy{[1..rk]};
 ConvertToVectorRepNC(newy, f);
 fi;
 return VectorSpaceToElement(geom1, newy * invchange);
 end;
 map := GeometryMorphismByFunction(ElementsOfIncidenceStructure(geom1),
 ElementsOfIncidenceStructure(geom2),
 func, false, pre );
 SetIsInjective( map, true );
 else
 Error("Polar spaces are not compatible");
 fi;
 return map;
end );

# CHECKED 28/09/11 jdb
# cmat comments: see NaturalEmbeddingBySubspace (above).
#############################################################################
#O NaturalEmbeddingBySubspaceNC( <ps1>, <ps2>, <v> )
## This operation is just like its namesake except that it
## has no checks
##
InstallMethod( NaturalEmbeddingBySubspaceNC,
"for a geometry into another, via a specified subspace, no-check version",
[ IsClassicalPolarSpace, IsClassicalPolarSpace, IsSubspaceOfProjectiveSpace ],
function( geom1, geom2, v )
 local map, rk, f, i, basis, invbasis, bs, c1, c2, orth, tyv, quad1, quad2,
 change, perp2, formonv, ses1, ses2, ty1, ty2, newmat, func, pre, invchange;
 rk := v!.type;
 ty1 := PolarSpaceType(geom1);
 ty2 := PolarSpaceType(geom2);
 f := geom1!.basefield;
 tyv := TypeOfSubspace( geom2, v );
 orth := ["parabolic", "elliptic", "hyperbolic"];
 if ty1 = ty2 or (ty1 in orth and ty2 in orth) then

 ## Let B be basis of v. Then the form BM(B^T)^sigma (n.b. sigma
 ## is a field automorphism), where M is the form for geom2,
 ## will be equivalent to the form for geom1.

 basis := Unpack(v!.obj);

 ## As usual we must consider two cases, the quadratic form case
 ## and the sesquilinear form case. We then obtain a base change matrix c1.

 if HasQuadraticForm( geom2 ) and HasQuadraticForm( geom1 ) then
 quad1 := QuadraticForm(geom1);
 quad2 := QuadraticForm(geom2);
 newmat := basis * quad2!.matrix * TransposedMat(basis);
 formonv := QuadraticFormByMatrix(newmat, f);
 c1 := BaseChangeToCanonical( quad1 );
 else
 ses1 := SesquilinearForm(geom1);
 ses2 := SesquilinearForm(geom2);
 if ty1 = "hermitian" then
 newmat := basis * ses2!.matrix * (TransposedMat(basis))^CompanionAutomorphism(geom1);
 formonv := HermitianFormByMatrix(newmat, f);
 else
 newmat := basis * ses2!.matrix * TransposedMat(basis);
 formonv := BilinearFormByMatrix(newmat, f);
 fi;
 c1 := BaseChangeToCanonical( ses1 );
 fi;

 ## Finding isometry from geom1 to polar space defined by formofv:

 c2 := BaseChangeToCanonical( formonv );
 change := c1^-1 * c2;
 #ConvertToMatrixRepNC(change, f);
 invchange := change^-1;
 bs := BaseSteinitzVectors(Basis(geom2!.vectorspace), basis);
 invbasis := Inverse(Concatenation(bs!.subspace, bs!.factorspace));
 #ConvertToMatrixRepNC(invbasis, f);

 func := x -> VectorSpaceToElement( geom2, Unpack(x!.obj) * change * basis);
 pre := function(y)
 local newy;
 newy:= Unpack(y!.obj) * invbasis;
 if IsMatrix(newy) then
 newy := newy{[1..Size(newy)]}{[1..rk]};
 ConvertToMatrixRepNC(newy, f);
 else
 newy := newy{[1..rk]};
 ConvertToVectorRepNC(newy, f);
 fi;
 return VectorSpaceToElement(geom1, newy * invchange);
 end;
 map := GeometryMorphismByFunction(ElementsOfIncidenceStructure(geom1),
 ElementsOfIncidenceStructure(geom2),
 ,false pre;
 SetIsInjective( map, true );
 else
 Error("Polar spaces are not compatible");
fi;
 return map;
end );

# CHECKED 28/09/<>infinity<C. Check whether for all generators <M>f</M>
#cmat changed 20/3/14.
#############################################################################
#O IsomorphismPolarSpaces( <ps1>, <ps2>, <bool> )
# returns the coordinate transformation from <ps1> to <ps2> (which must be
# polar spaces of the same type of course). If <bool> is true, then an intertwiner
# is computed.
##
InstallMethod( IsomorphismPolarSpaces,
 "for two similar polar spaces and a boolean",
 [ IsClassicalPolarSpace, IsClassicalPolarSpace, IsBool ],
function( ps1, ps2, computeintertwiner )
 local form1, form2, f, map, c1, c2, change, invchange, hom, ty1, ty2,
 coll1, coll2, gens1, gens2, x, y, func, inv, twinerfunc, twinerprefun, r1, r2;
 ty1 := PolarSpaceType( ps1 );
 ty2 := PolarSpaceType( ps2 );
 if ty1 = "parabolic" and ty2 = "symplectic" then
 return IsomorphismPolarSpacesProjectionFromNucleus(ps1, ps2, computeintertwiner);
 fi;
 r1 := Rank(ps1);
 r2 := Rank(ps2);
 if <>ty2orr1< r2 then
 Error("Polar spaces are of different type or of under <M>f</M> is finite. If not, return <C>infinity</C>.
 fi;
 f := ps1!.basefield;
 if f <> ps2!.basefield then
 2> must be polar overthe samefield")
 fi;
 if IsEvenInt(Size(f)) and ty1 in ["parabolic", "elliptic", "hyperbolic"] then
 form1 := QuadraticForm( ps1 );
 form2 := QuadraticForm( ps2 );
 else
 form1 := SesquilinearForm( ps1 );
 form2 := SesquilinearForm( ps2 );
fijava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
 c1 := BaseChangeToCanonical( form1 );
 c2 Apply the theball ofradius&bsp;
 change := NewMatrix(IsCMatRep, ps1!.basefield, ps1!.dimension+1,c1^- around&;1,and return<></C if .
 #ConvertToMatrixRep(change, Finally tocomputethe of ></>.
 invchange := Inverse(change);
 ty1=""then
 change := change^CompanionAutomorphism(ps2);
 invchange := invchange^CompanionAutomorphism(ps1);
 fi;
 func := function(x)
 return VectorSpaceToElement(ps2, ShallowCopy(x!.obj) * change);
 end;
 inv := function(x)
 return VectorSpaceToElement(ps1, ShallowCopy(x!.obj) * invchange);
 end;
 map := GeometryMorphismByFunction(ElementsOfIncidenceStructure(ps1),
 ElementsOfIncidenceStructure(ps2),
 func, inv);

 # map from gens2 to gens1
 twinerprefun := function( x )
 l y;
 y := MutableCopyMat( x!.mat );
 ProjElWithFrob(change *y ^x!.frob,x!frob, ;
 return ProjElWithFrob( change * y * invchange^(x<>asily/> ..by checking whether 1 Mg<M>

 end;

 # map from gens1 to gens2
 twinerfunc := function( java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 59
 local x;
 x := MutableCopyMat( y!.mat );
 #return ProjElWithFrob( invchange * x * change^y!.frob, y!.frob, f );
 return ProjElWithFrob( invchange * x * change^(y!.frob^-1), y!.frob, f );
 end;
 if computeintertwiner then
 if (HasIsCanonicalPolarSpace( ps1 ) and IsCanonicalPolarSpace( ps1 )) or
 /Item>
 coll1 := CollineationGroup(ps1);
 gens1 := GeneratorsOfGroup( coll1 );
 gens2 := List(gens1, twinerfunc);
 coll2 := GroupWithGenerators(gens2);
 hom := GroupHomomorphismByFunction(coll1, coll2, twinerfunc, twinerprefun);
 SetIntertwiner( map, hom );
 UseIsomorphismRelation(coll1,coll2);
 elif (HasIsCanonicalPolarSpace( ps2 ) and IsCanonicalPolarSpace( ps2 )) or
 HasCollineationGroup( ps2 ) then
 coll2 := CollineationGroup( ps2 );
 gens2 := GeneratorsOfGroup( coll2 );
 gens1 := java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 8
 coll1 =GroupWithGenerators(ens1)
 hom := GroupHomomorphismByFunction(coll1, java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 9
 SetIntertwiner( map, hom );
 UseIsomorphismRelation(coll1, the connected of the transition
 else
 Info(java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
 java.lang.StringIndexOutOfBoundsException: Range [6, 7) out of bounds for length 6
 fi;
 return map;
end );

# CHECKED 28/09/11 jdb
########################################
#O IsomorphismPolarSpaces( <ps1>, <ps2> )
# IsomorphismPolarSpaces(<s1> ps2>, ;
##
InstallMethod( IsomorphismPolarSpaces
"for twosimilar polar spaces"
[ IsClassicalPolarSpace, IsClassicalPolarSpace ],
function( ps1, ps2 )
 return IsomorphismPolarSpaces(ps1,ps2,true );
end );

#CHECKED28/11jdb
#######################################<Item>
#O IsomorphismPolarSpacesNC( <ps1>, <ps2>, <bool> )
# This operation is just like its namesake except that it
## has no checks
java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 0
 "for two similar polar spaces and a boolean",
[IsClassicalPolarSpaceIsClassicalPolarSpace,IsBool]java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
function( />
 local form1, form2, f, map, c1, c2, change, invchange, hom, ty1, ty2, n,
 coll1, coll2, <tem>
 ty1 := PolarSpaceType( ps1 );
 ty2 := PolarSpaceType( ps2 );
 f := ps1!.basefield;
 if IsEvenInt(Size(f)) and ty1 in ["parabolic", "elliptic", "java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 0
 form1 := QuadraticForm( ps1 G<A>)</>
 form2 := QuadraticForm( ps2 );
 else
 form1 := java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 9
 form2 := SesquilinearForm( ps2 );
 T methoduses acombinationoftechniquestoobtainsome basic
 c1 := BaseChangeToCanonical( form1 );
orm2 )java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
 , ps1.basefield,ps1!dimension1c1-1 *c2) #
 #ConvertToMatrixRepNC(change, f);
ange: (;java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
 func :
 return VectorSpaceToElement
 end;
 inv : function(x)
 return java.lang.StringIndexOutOfBoundsException: Range [0, 30) out of bounds for length 28
 end;
 map := GeometryMorphismByFunction(ElementsOfIncidenceStructureItem>
 java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 42

 # intertwiner.java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
# map from gens2 to gens1
 twinerprefun := function( x )
 local <>Trivial<.
 y := MutableCopyMat( x!.mat );
 #return ProjElWithFrob( invchange * x * change^y!.frob,
 #return java.lang.StringIndexOutOfBoundsException: Range [0, 38) out of bounds for length 8
 ProjElWithFrob( y*invchange(!frob^),x.frob, )
 end;

 # map from gens1 to gens2
 twinerfunc := function( y )
 local x;
 x := MutableCopyMat( y!.mat );
 #return ProjElWithFrob( invchange * x * change^y!.frob, y!<Item>
 return ProjElWithFrob( invchange *
 end;
 if computeintertwiner then
 if (asIsCanonicalPolarSpaceps1)andIsCanonicalPolarSpace(ps1 )) or
 HasCollineationGroup( ps1 ) then
 :CollineationGroup)java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
 gens1 := GeneratorsOfGroup( coll1 );
 gens2 : List(gens1,)
 coll2 := GroupWithGenerators(gens2);
 =GroupHomomorphismByFunction(oll1 ,twinerfunc twinerprefun)
 SetIntertwiner( map, hom );
 UseIsomorphismRelation(oll1,)
 elif (HasIsCanonicalPolarSpace( ps2 ) and java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 9
 HasCollineationGroup( ps2 ) then Markjava.lang.StringIndexOutOfBoundsException: Range [8, 9) out of bounds for length 8
 coll2 := CollineationGroup( ps2 );
ratorsOfGroup;
 gens1 := List(gens2, twinerprefun
 coll1 := GroupWithGenerators(gens1);
 hom := GroupHomomorphismByFunction(coll1<ark>
 SetIntertwiner( map, hom );
 UseIsomorphismRelation(coll1,coll2);
 else
 (,1 "No intertwiner computed. One of the groupcomputed)java.lang.StringIndexOutOfBoundsException: Index 116 out of bounds for length 116
 fi;
 fi;
 java.lang.StringIndexOutOfBoundsException: Range [2, 1) out of bounds for length 9
end );

# CHECKED 28/09/11 jdb
#############################################################################
#O IsomorphismPolarSpacesNC( <ps1>, <ps2> )
# returns IsomorphismPolarSpacesNC( <ps1>, <ps2>, true );
##
InstallMethod( IsomorphismPolarSpacesNC,
"for two similar polar spaces",
sClassicalPolarSpace,IsClassicalPolarSpace ],
function( ps1, ps2 )
 return IsomorphismPolarSpacesNC( ps1, ps2, true );
end );

###############################Uses <CDirectProduct<C>to embedthe <C>NrMovedPoints(<A>P</A>)</C>th
## Field reduction / subfield morphisms. Helper operations.
##########################################################

# CHECKED 28/09/11 jdb
#############################################################################
# <B,<> )
# returns the preimage of BlownUpMat
##
InstallMethod ShrinkMat,
"for a basis and a matrix",
[ IsBasis, IsMatrix ],
function

 #THERE WASA BUGINTHIS FUNCTION lavrauw/2010
 ## NOT ALL MATRICES ARE BLOWNUP MATRICES! Here is a new version<C>(<A></><><A><Cjava.lang.StringIndexOutOfBoundsException: Range [43, 44) out of bounds for length 43
 #Here is the explanation thatcouldgointo documentation:
 ## The result of BlownUpMat(B,A) is the Restricts AG<A> the residue &;() and the
 #matrix M_a the linear |- axwith tothe B of
 ## the field K seen \tau_{0(2),1(4)}</M> is usedthat the imagesof ()under
 ## B={b_1,b_2,...,b_d}, then powers of this mapping are paiof this pairwisedisjoint residueclassesjava.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
 ## Concersely, if we
 ## a blown up matrix.
 ## This can be done as follows. Let A be a<>(A><A)/C>
 ## we need to check that the set /Markjava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
 ## (Since this would be the a \in K <>rivial/Q>

 ## This function is basically the inverse
 ## BlownUpMat.

local d,vecs,m,n,bmat,blocks,bl,submat,i
d:=Size(B);
vecs:=BasisVectors(B);
m
n:=NrCols(mat);
 # First we check if m and n are multiples of d
if not (IsInt(m/d) and IsInt(n/d)) then
 Error("The matrix does not have the right dimensions");
fi;
blocks:=List(Cartesian([0..m/d-1],[0..n/d-1]),ij->mat{[ij[1]*d+1 .. ij[1]*d+d]}{[ij[2]*d+1 ..ij[2]*d+d]});
newmat:=[];
for i in [1..m/d] do
 row:=[];
 for j in [1..n/d] do
 #submat:=blocks[(i-1)*d+j]; #wrong line?
 submat:=blocks[(i-1)*(m/d)+j];
 checklist:=List([1..d],i->(vecs[i]^(-1)*(Sum([1..d],j->submat[i][j]*vecs[j]))));
 if Size(AsSet(checklist))<>1 then
 Error("The matrix is not a blown up matrix");
 fi;
 Add(row,checklist[1]);
 od;
 Add(newmat,row);
od;
return newmat;
 # DO WE NEED TO USE ConvertToMatrixRepNC ?
end);


InstallMethod( ShrinkMat,
"for a field, a subfield and a matrix",
[ IsField,IsField, IsMatrix ],
function( f1,f2,mat )
# This gives the same result as ShrinkMat(B,mat) with B the natural basis returned by
# Basis(AsVectorSpace(f2,f1));
local B;
B:=Basis(AsVectorSpace(f2,f1));
return ShrinkMat(B,mat);
end );

#############################################################################
#O ShrinkVec( <f1>, <f2>, <v> )
# f2 is a subfield of f1 and v is vector in a vectorspace V2 over f2
# return the vector of length d2/t, where d2=dim(V2), and t=[f1:f2]
# using the natural basis Basis(AsVectorSpace(f2,f1))
##
InstallMethod( ShrinkVec,
"for a field, a subfield and a vector",
[ IsField, IsField, IsVector ],
function( f1,f2,v )
local t,d1,d2,basvecs;
t:=Dimension(AsVectorSpace(f2,f1));
d1:=Length(v)/t;
if IsInt(d1) then
 basvecs:=BasisVectors(Basis(AsVectorSpace(f2,f1)));
 return List([1..d1],i->v{[(i-1)*t+1..i*t]}*basvecs);
else
 Error("The length of v is not divisible by the degree of the field extension");
fi;
end );

#############################################################################
#O ShrinkVec( <f1>, <f2>, <v>, basis )
# The same operation as above, but now with a basis as extra argument
InstallMethod( ShrinkVec,
"for a field, a subfield and a vector",
[ IsField, IsField, IsVector, IsBasis ],
function( f1,f2,v,basis )
localt,d2,;
t:=Dimension(AsVectorSpace(f2,f1));
d1:=Length(v)/t;
if IsInt(d1) hen
 basvecs# iscomputed.
 return List([1InstallMethod IsomorphismPolarSpaces,java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
else
 Error (ps1, ps2,computeintertwiner )
fi;
end );

#######################ty1 : PolarSpaceType(ps1 );
#O BlownUpSubspaceOfProjectiveSpace( , <pg1> )
# if ty1parabolic = ""then
java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 2
java.lang.StringIndexOutOfBoundsException: Range [8, 5) out of bounds for length 24
"for a change := C(ps2;
[ IsBasis, IsProjectiveSpace ],
function invchange: invchange^ompanionAutomorphismps1)
 ,t,,g2m,mat2java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
q=!.;
 t:=basis!.d;
 if not if not
 Error" the space not java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 29

 r:=Dimension(pg1)+1;
 pg2:= ProjElWithFrob invchange*x*change(!.frob-), y!frob,f)
c,;
end ) lif (( ps2 ) and IsCanonicalPolarSpace)) or

############################ coll2: (ps2 );
#O java.lang.StringIndexOutOfBoundsException: Range [38, 36) out of bounds for length 57
# blowsup projective by field reduction..t.thecanonicalbasis
##
nstallMethod BlownUpProjectiveSpaceBySubfield
" SetIntertwiner(map,hom IsomorphismPolarSpaces ps1,<ps2, );
[ IsField, IsProjectiveSpace ],
function(subfield,pg)
local t,r,field;
 field:=LeftActingDomain(UnderlyingVectorSpace(pg))[IsClassicalPolarSpace IsClassicalPolarSpace ,
 if not subfield in Subfields(field) then
 Error("The first argument is not a subfield of the basefield of the projective space!");
 fi;
 t:=Dimension(AsVectorSpace(subfield,field));
 r:=Dimension(pg)+1;
 return(subfield);
 )

# CHECKED 1/10/11 jdb
#######################################
#O java.lang.StringIndexOutOfBoundsException: Range [39, 36) out of bounds for length 56
# blows ##has checks
##
InstallMethod( "for two similar polar spaces and,
"for a basis and a subspace of a projective space",
[,IsSubspaceOfProjectiveSpace,
functionps1 ,computeintertwiner java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
local pg1,q,t,r,pg2,mat1,mat2;

 q:=basis!.q;
 t:=basis!.d;
 =GF(^) then
 (The basisandthesubspaceare notcompatible")java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
 ;
 r:=Dimension(pg1)+1;
 pg2:=PG(r*t-1 form2 :SesquilinearForm )
 c =BaseChangeToCanonical form1 ;
 !. =1 then
 mat1 := [change := NewMatrix(IsCMatRep, ps1!.basefield, ps1!.dimension+1,c1^-1 * c2); #cmat change here
 fi;
 mat2:BlownUpMat(asismat1)java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
 return VectorSpaceToElement(pg2,mat2);
end );

#############################################################################
#O func = function()
# blows up a subspace of projective space by field reduction
#Thisis...to the of over thejava.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71
java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 2
InstallMethod BlownUpSubspaceOfProjectiveSpaceBySubfield
"end;
[ ],
function,)
local pg,field,basis;
 pg:=AmbientGeometry(subspace);
 field:=LeftActingDomain(UnderlyingVectorSpace(pg));
 if not subfield in Subfields(field) then
 Error" first argument is not a subfield the basefield space!);

basis=((,field))
 return BlownUpSubspaceOfProjectiveSpace(basis,subspace y :=MutableCopyMat(x. )
end );

###### #return ProjElWithFrob x *change(!frob^1) y!frob,f
#O return ProjElWithFrob( change * y * invchange^(x!.frob^-1), x!.frob, f );
# checks if a subspace is a blown java.lang.StringIndexOutOfBoundsException: Range [0, 36) out of bounds for length 7
##
InstallMethod( IsDesarguesianSpreadElement x;
" of aprojective space"
[ IsBasis, IsSubspaceOfProjectiveSpace ],
function(basis,subspace)
 local flag,q,t,pg1,pg2,em,basvecs,v,v1,i,mat,rt,r;
 flag:=true;
 q:=basis!.q;
 t:= return ProjElWithFrob(invchange*x * change^!frob,.,f );
 pg2:=AmbientGeometry(subspace);
 rt:=Dimension(pg2)+1;
 if java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 6
 if(HasIsCanonicalPolarSpace(ps1 and ( ps1 )or
else
 :rtt
 :PG(-,q^)
 basvecs gens2:List(ens1,)
 v:=Coordinates(Random(Points(subspace)));
 :List(.]i>{[i-1)t+.i*t}basvecs)
 mat:=BlownUpMat(basis,[v1 hom :=GroupHomomorphismByFunction( coll2 ,twinerprefun;
 :subspace VectorSpaceToElement(g2mat)
 fi;
 return flag;
end )

# To be done: check this method! (but it is currently never used).
#############################################################java.lang.StringIndexOutOfBoundsException: Range [26, 24) out of bounds for length 69
#O IsBlownUpSubspaceOfProjectiveSpace : java.lang.StringIndexOutOfBoundsException: Range [32, 30) out of bounds for length 50
# checks if := ()
# hom =GroupHomomorphismByFunctioncoll1,coll2,twinerfunc twinerprefun;
SetIntert(map ) java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
" else
[ IsBasis, IsSubspaceOfProjectiveSpace ],
# It Info(nfoFinInG 1,No intertwinercomputed ofthe musthaveacollineation computed)java.lang.StringIndexOutOfBoundsException: Index 116 out of bounds for length 116
# since for every subspace, provided################################################
#withrespecttowhichthe subspace is .
function(# ( <ps1> ps2> true)
 local flag,q,F,t,K,pg2,
=;
 q:=basis!.q;
 F:=GF(q);
 :=basis!.;
;
 end)
 rt:=Dimension(pg2)+1;
 if not (IsInt(rt/t) and IsInt((Dimension(subspace)+1)/t)) then
 flag:=false;
 else
 r:=rt/t;
 pg1:=PG(r-1,q^t);
 basvecs:=BasisVectors(basis);
 mat1:=[];
 span:=[];
 repeat
 repeat x:##############################java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
 ###############################
 v1:=List([1..r],i->v{[(i-1)*t+1..i*t]}*basvecs);
 Add(mat1,v1);
 span:=VectorSpaceToElement(pg2,BlownUpMat(basis,mat1));
 CHECKED 28/9/ jdb
 if not span##############################################
 : ;
 fi;
 fi;
return flag;
end );

########################################
# PROJECTIVE [ IsBasis, IsMatrix,
#########################################java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77

#
# recent onfieldreductionofprojective is
# Here thatcould go into documentation:
# The result (,A isthe whereeach aA_ijis

#######################
#O # thefieldK seen asddimensional overF This means thatif
geom2>is a projectivespaceovera field,<>is a projective
# over a field,if we implement ,we needtocheckif input
 <geom1>and <geom2>have the same ambientvectorspaceover K,then this
# ## This be done as .LetAbe a dm dn.Foreachdxd block, say
#ofLoverK.
# Importantnote: the intertwiner assourcethe Projectivitygroup* only,not
# the
# ##Thisfunctionis theinversemapof
InstallMethod( NaturalEmbeddingByFieldReduction,
"for projectivespaces and a basis",
 [ IsProjectiveSpace, IsField, IsBasis ],
function(pg1,f2,basis )

 :BasisVectorsB);

 local map:N();
 : pg1n)
q1: Sizef)java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
q =Size(2;
 t := LogInt(q1,q2);
 if not q2^t = q1 then
 Error( "<f2> is not a subfield for in [1../d]do
 fi;
 d1 row:[;
 d2=d1*t;
 pg2 := ProjectiveSpace(d2-1,q2);
 I( =GF((!q^!d)andf2=F(asis.)and d1*(asis!d)=d2)then
 Error("The basis submat:blocks[i1*m/)j];
f;

fun : function(subspace ) #This blowsup a subspaceof geom1 to a subspace of geom2
 local i
 return (basis,)java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
 mat1 o;
 if subspace!.type = 1#DO WENEED TO USE ConvertToMatrixRepNC?
 mat1 := [subspace!.obj];
 fi;
 java.lang.StringIndexOutOfBoundsException: Range [2, 1) out of bounds for length 2
 java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 50
 end IsField,,IsMatrix ],

 prefun := function#Thisgives the same as (,mat)withBthe
 ,asvecsspanxvv1,vecs;
 flag:=true;
 if not subspace in pg2local B;
 " fieldreduction !)
 ;
 if )
 flag := false;
 #########################################
 =()
 (UnderlyingObject);
 mat1 := List(vecs,x->List([1..d1],i->x{[(i-1#return the vector oflength d2/,where =imV) =f1:2
 : VectorSpaceToElementp,lownUpMat(basis,);
 if not span = subspace I( ShrinkVec
 flag := false;
 fi;
 fi;
 if flag= falsefunction(f1,,)
 Error" is in of field reductionmap!)java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83
 ;
 return VectorSpaceToElement(pg1,mat1);
 end;

 #prefun := function( x ) #I want to find out why this is not correct...
 # return VectorSpaceToElement(pg1,ShrinkMat(basis,x!.obj));
 #end;

 map := GeometryMorphismByFunction(ElementsOfIncidenceStructure(pg1),
 java.lang.StringIndexOutOfBoundsException: Range [70, 69) out of bounds for length 75
false java.lang.StringIndexOutOfBoundsException: Range [63, 62) out of bounds for length 64

 SetIsInjective( map, ############################################################

 # creatingintertwiner
 #126/:added see whether isreallya
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
 hom := function( m )
 imagejava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
 if ,d1,basvecs;
 fail;
 (InfoFinInG 1,"< not a ";
 fi;
 image := BlownUpMat(basis, m!.mat);
 #ConvertToMatrixRepNC=();
lineationOfProjectiveSpace, f2)java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51

 hominv := function( m )
 local preimage;
 if not IsOne(m!.frob) then
 return fail;
 Info(InfoFinInG, 1, "<el> is not a projectivity");
 fi;
 preimage := ShrinkMat(basis, Unpack(#################################################
 #ConvertToMatrixRepNCOBlownUpSubspaceOfProjectiveSpaceB,<pg1>)
 return ollineationOfProjectiveSpace,f1;
 #

: ProjectivityGrouppg1 )java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
 gensfunction(,g1)
 newgens := List(gens, hom);
 g2 := Group newgens )
 SetSize(,Size(1);
 twiner=!.;
 SetIntertwinerjava.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 37
 return map;
end );

######################## return;
#O NaturalEmbeddingByFieldReduction( <geom1>, <field> )
#field>is a field K <>is aprojective
# over a field extensionO BlownUpProjectiveSpaceBySubfield<>> )
#thata space geom2 sameambient overK.
# operation#
# L over K.
##
InstallMethod NaturalEmbeddingByFieldReduction
"for a projective space and a field",
[ IsProjectiveSpace, IsField ],
function( pg1,local t,,ield;
 local basis;
 basis:=Basis(AsVectorSpace(f2,pg1!.basefield));
 returnNaturalEmbeddingByFieldReductionpg1,2,asis)
nd)

##t:DimensionAsVectorSpaces,));
O ( <,<> basis> )
# <geom2> is a projective return PG(r*t-1,Size);
overa field L, and considering L asa vectorspaceover K, yields
# java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0
########################################
# of L over K.
##
InstallMethod#
 a basis"
[ r basis a subspaceof projective space",
[IsBasis ],
 return NaturalEmbeddingByFieldReduction(pg1,pg2!.basefield,basis);
end);

# CHECKED 28/09/11 jdb
#################q=basis!.;
#ONaturalEmbeddingByFieldReduction geom1> <geom2>)
#<isa projective space over a field K geom1>is projective space
# over a field extension L, and considering L as a vector space over K, yields
# that <geom1> and <geom2> have the same ambient vectorspace over K, then fi;
# operation returns the natural pg2:PG(*1q;
# L over K.
:subspace.obj;
nstallMethod ,
ortwo projective spaces"
[ java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 6

 local basis;
 basis:=Basis##########################################
 return BlownUpSubspaceOfProjectiveSpaceBySubfield( <subfield>, <subspace> )
end );

#######################################
 SPACES
###################################

#
# Two references ifnotsubfield ( then
rouptheoreticpointofview Polarspacesandembeddings "
# (New Zealand J. fi;
nite point of view F reductionandlinearsetsinfinitegeomety Lavrauw
# and return(basis,)
#
# We will use [2] as##########################################

java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 4
#
# First the field reduction of bilinear forms, quadratic forms and hermitian forms
#
#####

#################################ocal flagqt,pg1,,m,,v,v1,,at,rt,;
#O BilinearFormFieldReduction( <bil1>, <f2>, <alpha>, <basis> )
# q:basis!q;
# <alpha> determines the linear map from f1 pg2:AmbientGeometry)java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
# basis.This then
#
##
# (l31/10/13)The fourthargument a over K. his just
# basvecs=()
# v:=Coordinates(Random(Points(subspace)));
# There is also a version of this field v1:List(.ri>(-1*+.it]*)
# a basis as mat:BlownUpMatbasis,v1)

InstallMethod( BilinearFormFieldReduction,
"for a bilinear form and a ;
[ IsBilinearForm, IsField, IsFFE, java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 7
function(bil1,f2,alpha,basis)
# f2 is a subfield of the basefield of ##############################################################
ocal ,,,,d2V2,V1p,b2vecsmat2ij;
 checksif a subspace is blown field reduction,w...a basis
f1:=bil1!.basefield;
d1:=NrRows(mat1);
t=imensionAsVectorSpacef2f));
d2:=d1*t;
V2:=f2^d2;
V1:=f1^d1;
b2:=Basis(V2);
b2vecs:=BasisVectors(b2);
mat2:=[];
for i in [1..d2] do
 row:=[];
 .d2]do
 Add(,,F,,,pg2,rt,,g1,asvecsm,panxv,v1,;
 od;
 Add(mat2,row);
od;
bil2:=BilinearFormByMatrix(mat2, K:(q)
return bil2java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
end );

################################ r=/;
# BilinearFormFieldReduction(<
# Thebasvecs:BasisVectorsjava.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 32
##
InstallMethod( repeat x:=Random(Points(subspace)) in span
"for a bilinear form and a field",
 , IsField, IsFFE]
function(bil1,f2,alpha)
# f2 :VectorSpaceToElement(pg2B(mat1);
return BilinearFormFieldReduction(bil1,f2,alpha,Basis(AsVectorSpace(f2,bil1!.basefield until (=(subspace)
end )

######## ;
#O QuadraticFormFieldReduction( <qf1>, <f2> return flag;
# <bil1> is a bilinear form java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 2
# returns the bilinear java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 20
##
# REMARK (java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
# extra functionality to the user, and it is used as# Vande ( 2013java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
ShrinkVec
field reductionoperation,which does not
# a # <geom2> projective overafieldK <eom1> projective space

InstallMethod( QuadraticFormFieldReduction,
fora quadraticformanda field,
[ operation the naturalembedding .e with relationto thegiven basis
function(qf1,f2,alpha,basis)
 isa subfield the for quadratic
local f1,basvecs, the full collineation !
f1:=qf1!.basefield;
basvecs:=BasisVectors(basis);
d1:=NrRows(qf1!.matrix);
t:=Dimension(AsVectorSpace(f2,f1));
d2:=d1*t;
V2:=f2^d2;
V1:=f1^d1;
b2:=Basis(V2);
b2vecs:=BasisVectors(b2);
mat2:=IdentityMat(d2,f2);
for i in [1..d2] do
 mat2[i,i]:=Trace(f1,f2,alpha*((ShrinkVec(f1,f2,b2vecs[i],basis))^qf1));
od;
for i in [1..d2-1] do
 for j in [i+1..d2] do
 mat2[i,j]:=Trace(f1,f2,alpha*((ShrinkVec(f1,f2,b2vecs[i]+b2vecs[j],basis))^qf1
 -(ShrinkVec(f1,f2,b2vecs[i],basis))^qf1-(ShrinkVec(f1,f2,b2vecs[j],basis))^qf1));
 #mat2[j,i]:=mat2[i,j]; THESE entries need to be zero
 od;
od;
qf2:=QuadraticFormByMatrix(mat2,f2);
return qf2;
end );

#############################################################################
# #O QuadraticFormFieldReduction( <qf1>, <f2>, <alpha> )
# The same as above, but without specified basis. In this case the canonical basis is used.
##
InstallMethod( QuadraticFormFieldReduction,
"for a quadratic form and a field",
[ IsQuadraticForm, IsField, IsFFE ],
function(qf1,f2,alpha)
local basis,qf2;
# f2 is a subfield of the basefield for the quadratic form q1
basis:=Basis(AsVectorSpace(f2,qf1!.basefield));
qf2:=QuadraticFormFieldReduction(qf1,f2,alpha,basis);
return qf2;
end );

#############################################################################
#O HermitianFormFieldReduction( <hf1>, <f2>, <alpha>, <basis> )
# <hf1> is a hermitian form over <f1>, <f2> is a subfield of <f1>. This operation
# returns the form T(alpha*bil1(.,.)). Depending on the degree of the field extension, this
# yields either a bilinear form or a hermitian form.
##
# REMARK (ml 31/10/13) The fourth argument is a basis of L over K. This just gives
# extra functionality to the user, and it is used as third argument in the
# operation ShrinkVec.
# There is also a version of this field reduction operation, which does not use
# a basis as fourth argument. In this case the standard GAP basis is used.

InstallMethod( HermitianFormFieldReduction,
"for a hermitian form and a field",
[ IsHermitianForm, IsField, IsFFE, IsBasis ],
function(hf1,f2,alpha,basis)
# f2 is a subfield of the basefield for the hermitian form hf1
# here the basefield is always a square
local f1,d1,t,d2,V2,V1,phi,b2,b2vecs,mat2,i,row,j,hf2;
f1:=hf1!.basefield;
d1:=NrRows(hf1!.matrix);
t:=Dimension(AsVectorSpace(f2,f1));
d2:=d1*t;
V2:=f2^d2;
V1:=f1^d1;
b2:=Basis(V2);
b2vecs:=BasisVectors(b2);
mat2:=[];
for i in [1..d2] do
 row:=[];
 for j in [1..d2] do
 Add(row,Trace(f1,f2,alpha*([ShrinkVec(f1,f2,b2vecs[i],basis),ShrinkVec(f1,f2,b2vecs[j],basis)]^hf1)));
 od;
 Add(mat2,row);
od;
# checking parity of t is indeed sufficient to decide, see table in manual.
if IsOddInt(t) then
 hf2:=HermitianFormByMatrix(mat2,f2);
else
 hf2:=BilinearFormByMatrix(mat2,f2);
fi;
return hf2;
end );

#############################################################################
# #O HermitianFormFieldReduction( <hf11>, <f2>, <alpha> )
# The same as above, but without specified basis. In this case the canonical basis is used.
##

InstallMethod( HermitianFormFieldReduction,
"for a hermitian form and a field",
[ IsHermitianForm, IsField, IsFFE ],
function(hf1,f2,alpha)
# f2 is a subfield of the basefield for the hermitian form hf1
# here the basefield is always a square
local basis;

basis:=Basis(AsVectorSpace(f2,hf1!.basefield));
return HermitianFormFieldReduction(hf1,f2,alpha,basis);
end );

##########################################################
#
# Next the embeddings for polar spaces based on the field reduction of the forms above
#
#############################################################################

# master version for the user: handle all parameters.
#############################################################################
#O NaturalEmbeddingByFieldReduction( <ps1>, <f2>, <alpha>, <basis>, <bool> )
# returns a geometry morphism, described below. If <bool> is true, then an intertwiner
# is computed.
##
InstallMethod (NaturalEmbeddingByFieldReduction,
"for a polar space, a field, a basis, a finite field element, and a boolean",
[IsClassicalPolarSpace, IsField, IsFFE, IsBasis, IsBool],
function(ps1,f2,alpha,basis,computeintertwiner)
# f2 must be a subfield of the basefield of the classical polar space
local type1,qf1,qf2,ps2,bil1,bil2,hf1,hf2,fun,prefun,f1,
 map,hom,hominv,g1,gens,newgens,g2,twiner, t,q1,q2,d1,d2;
type1 := PolarSpaceType(ps1);
f1:=ps1!.basefield;
q1 := Size(f1);
q2 := Size(f2);
t := LogInt(q1,q2);
d1 := ps1!.dimension + 1;
d2 := d1*t;

# 1. the polar space is of orthogonal type
if type1 in ["hyperbolic","elliptic","parabolic"] then
 qf1:=QuadraticForm(ps1);
 qf2:=QuadraticFormFieldReduction(qf1,f2,alpha,basis);

 if IsSingularForm(qf2) then
 Error("The field reduction does not yield a natural embedding");
 else ps2:=PolarSpace(qf2);
 fi;
# 2. the polar space is of symplectic type
elif type1 in ["symplectic"] then
 bil1:=SesquilinearForm(ps1);
 bil2:=BilinearFormFieldReduction(bil1,f2,alpha,basis);
 ps2:=PolarSpace(bil2);
# 3. the polar space is of hermitian type
elif type1 in ["hermitian"] then
 hf1:=SesquilinearForm(ps1);
 hf2:=HermitianFormFieldReduction(hf1,f2,alpha,basis);
 ps2:=PolarSpace(hf2);
fi;

#em:=NaturalEmbeddingByFieldReduction(AmbientSpace(ps1),AmbientSpace(ps2));
# this is the field reduction for projective spaces PG(r-1,q^t) -> PG(rt-1,q)

#fun:=function(x)
# local projfun;
# projfun:=em!.fun;
# return VectorSpaceToElement(ps2,projfun(x)!.obj);
#end;

 fun := function( subspace )
 local mat1,mat2;
 mat1 := subspace!.obj;
 fsubspace!type = 1 then
 mat1 := [subspace!.obj];
 fi;
 mat2 := BlownUpMat(basis,mat1);
 return VectorSpaceToElement(ps2,mat2);
 end;

#prefun:=function(x)
# local projprefun;
# projprefun:=em!.prefun;
# return VectorSpaceToElement(ps1,projprefun(x)!.obj);
#end;

 # new version (2/ function( pg1 f2 basis )
 prefun :# Thismorphism contains func and prefunc with built- checkjava.lang.StringIndexOutOfBoundsException: Range [67, 68) out of bounds for length 67
 flagbasvecsmat1spanxvv1,,vecsjava.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
 flag:=true;
 if not subspace in ps2 then
 Error("The input is not in the range of the field reduction map!");
 fit: LogInt(,)java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
 if not IsInt((Dimension)+/t)then

 else
 =()
 mat1:=[];
 :[;
 vecs := Unpack(UnderlyingObject(subspace)) i
 mat1 := List( fun : function(subspace)#This map blows up a subspace of geom1 to a subspace of geom2
 span : local mat1,java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
 := subspace!.obj;
 if not span=subspace then
 flag := false;
 fi;

 if flag= false then
 range of the field reduction map)java.lang.StringIndexOutOfBoundsException: Range [79, 80) out of bounds for length 79
 fi;
 VectorSpaceToElement,);
 end;

m : GeometryMorphismByFunctionElementsOfIncidenceStructure), (ps2,
 fun,flag:=;

java.lang.StringIndexOutOfBoundsException: Range [16, 1) out of bounds for length 83

## Now creating if not IsIntDimension(ubspace+)t

java.lang.StringIndexOutOfBoundsException: Range [12, 3) out of bounds for length 16
 if (vecs := Unpack(UnderlyingObject))java.lang.StringIndexOutOfBoundsException: Range [59, 60) out of bounds for length 59
 ( ps1 ) then

 :=function )
 if span=subspacethen
 image := BlownUpMat(basis, m!.mat);
 ConvertToMatrixRepNC;
 return CollineationOfProjectiveSpace(image, f2);
 end;
 hominv := function( m )
 preimagejava.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
 :=(basis,Unpack(!mat);
 ConvertToMatrixRepNC( preimage, f1 );
 CollineationOfProjectiveSpacepreimage,f1);
 end;
 # return Vec(,hrinkMatbasis,x!));
 gens := GeneratorsOfGroup( g1 );
 newgens := List(gens, hom);
 g2 := Group( newgens );
 SetSize(g2, Size(g1));
 : g1 g2,hom,hominv)
 SetIntertwiner( map, SetIsInjective ,true )
 fi;
fi;

 return map;
 hom :=function( )

#first version: user wants a particular alpha, and basis, agrees with bool fail;
################################################################fi;
# NaturalEmbeddingByFieldReduction(<> <> <alpha> <>)
# returns NaturalEmbeddingByFieldReduction( <java.lang.StringIndexOutOfBoundsException: Range [0, 50) out of bounds for length 38
#
InstallMethod java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
" local preimage;
[IsClassicalPolarSpace, IsField, IsFFE, IsBasis],
function(ps1,f2,alpha,basis)
 return NaturalEmbeddingByFieldReduction(ps1,f2,alpha,basis,ty"java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
);

#java.lang.StringIndexOutOfBoundsException: Range [18, 7) out of bounds for length 87
############### return CollineationOfProjectiveSpace(,f1);
#O NaturalEmbeddingByFieldReduction( <geom1>, <f2>, <alpha>, <java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 0
# gens: (g1);
 < is thecanonical basisof BaseField(eom1 <>
java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
InstallMethod (NaturalEmbeddingByFieldReduction,
"for a polar space and a field, a finite field element, and SetIntertwiner(map,twiner);
[IsClassicalPolarSpace, IsField, java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 7
function(,f2,lphabool)
 return NaturalEmbeddingByFieldReduction(s1f2,,(f,(ps1)),bool)
end );

#third particular version: user wants#<> isa K,<geom1>is a projective space
#############################################################################
#O NaturalEmbeddingByFieldReduction( <geom1>, <f2>, <alpha> )
# returns NaturalEmbeddingByFieldReduction( <geom1>, <f2>, <alpha>, <basis>, true )
# where <basis> is the canonical basis of BaseField(geom1) over <f2>.
#
InstallMethod (NaturalEmbeddingByFieldReduction,
"for a polar space, a field, and a finite field element",
[IsClassicalPolarSpace, IsField, IsFFE],
function(ps1,f2,alpha)
 return NaturalEmbeddingByFieldReduction(ps1,f2,alpha,Basis(AsVectorSpace(f2,BaseField(ps1))),true);
end );

# fourth particular version: user agrees but wants to control intertwiner
#############################################################################
#O NaturalEmbeddingByFieldReduction( <geom1>, <f2>, <bool> )
# returns NaturalEmbeddingByFieldReduction( <geom1>, <f2>, <one>, <basis>, <bool> )
# where <basis> is the canonical basis of BaseField(geom1) over <f2>, and
# <one> is One(BaseField(geom1)). This might be usefull if you agree with the defaults, but doesn't
# want the intertwiner.
#
InstallMethod (NaturalEmbeddingByFieldReduction,
"for a polar space, a field, and a boolean",
[IsClassicalPolarSpace, IsField, IsBool],
function(ps1,f2,bool)
 return NaturalEmbeddingByFieldReduction(ps1,f2,One(f2),Basis(AsVectorSpace(f2,BaseField(ps1))),bool);
end );

# fifth particular version: user agrees with everything
#############################################################################
#O NaturalEmbeddingByFieldReduction( <geom1>, <f2> )
# returns NaturalEmbeddingByFieldReduction( <geom1>, <f2>, <alpha>, <basis>, true )
# where <basis> is the canonical basis of BaseField(geom1) over <f2>, and
# <alpha> is One(f2).
InstallMethod (NaturalEmbeddingByFieldReduction,
"for a polar space and a field",
[IsClassicalPolarSpace, IsField],
function(ps1,f2)
 return NaturalEmbeddingByFieldReduction(ps1,f2,One(f2),Basis(AsVectorSpace(f2,BaseField(ps1))),true);
end );

# CHECKED 28/09/11 jdb
#############################################################################
#O NaturalEmbeddingByFieldReduction( <geom1>, <geom2> )
# returns NaturalEmbeddingByFieldReduction(<geom1>, <geom2> )
##
InstallMethod( NaturalEmbeddingByFieldReduction,
"for two polar spaces",
[ IsClassicalPolarSpace, IsClassicalPolarSpace ],
function( geom1, geom2 )
 return NaturalEmbeddingByFieldReduction( geom1, geom2, true );
end );

# ml 31/10/2013
# changed with new prefun 2/7/14 jdb
#####################################################################################
# O NaturalEmbeddingByFieldReduction
# This NaturalEmbeddingByFieldReduction takes two polar spaces and returns an embedding
# obtained by field reduction if such an embedding exists, otherwise it returns an error.
##
InstallMethod (NaturalEmbeddingByFieldReduction,
"for two classical polar spaces",
[IsClassicalPolarSpace, IsClassicalPolarSpace, IsBool],
function(ps1, ps2, computeintertwiner)

#####################################################################
# List of possible embeddings of ps1 in PG(r-1, q^t) -> ps2 in PG(rt-1, q)
#
# Hermitian
# H -> H (t odd)
# H -> W (t even)
# H -> Q+ (t even, r even)
# H -> Q- (t even, r odd)
#
# Symplectic
# W -> W (always)
#
# Orthogonal
# r even
# Q+ -> Q+ (always)
# Q- -> Q- (always)
# r odd
# Q -> Q (t odd, q odd)
# Q -> Q+ (t even, q odd)
# Q -> Q- (t even, q odd)
#####################################################################

 local t, r, type1, type2, form1, f1, f2, newps, sigma, map,
 gamma, bil1, q, n, alpha, basis, is_square, iso, form2, fun, prefun,
 c1, c2, cps2, cps2inv, d1, hom, hominv, g1, g2, gens, newgens, twiner;

type1 := PolarSpaceType(ps1);
type2 := PolarSpaceType(ps2);
if type1 in ["hyperbolic", "parabolic", "elliptic"] then
 form1 := QuadraticForm(ps1);
else
 form1 := SesquilinearForm(ps1);
fi;
 d1 := ps1!.dimension+1;
f1 := ps1!.basefield;
f2 := ps2!.basefield;
q := Size(f2);

if not IsSubset(f1,f2) then
 Error("Fields are incompatible");
fi;
t := Log(Size(f1),q);
r := ProjectiveDimension(ps1)+1;
basis := CanonicalBasis(AsVectorSpace(f2,f1));
 is_square := function(x, f) return IsZero(x) or IsEvenInt(LogFFE(x,PrimitiveRoot(f))); end;

if IsSesquilinearForm(form1) then
 if type1 = "hermitian" and type2 = "hermitian" and IsOddInt(t) then
 Info(InfoFinInG, 1, "These polar spaces are suitable for field reduction");
 # need alpha to be fixed by sigma:
 alpha := One(f1);
 #map := NaturalEmbeddingByFieldReduction( ps1, f2, alpha, basis );
 form2 := HermitianFormFieldReduction(form1,f2,alpha,basis);
 elif type1 = "hermitian" and type2 = "symplectic" and IsEvenInt(t) then
 Info(InfoFinInG, 1, "These polar spaces are suitable for field reduction");
 if IsEvenInt(q) then
 # need alpha to be fixed by sigma:
 alpha := One(f1);
 else
 # need sigma(alpha)=-alpha
 sigma := Sqrt(Size(f1));
 alpha := First over afield extensionL,and considering L a vector space overK yields
 fi;
 #map := NaturalEmbeddingByFieldReduction( ps1, f2, alpha, basis );
 form2 := HermitianFormFieldReduction(form1,f2,alpha,basis);

 elif InstallMethod( ,
 IsOddIntq)
 Info(InfoFinInG, 1, "[ IsProjectiveSpace, IsField, ]
 alpha := One(f1);
 #map basis:=asisAsVectorSpace(2,!basefield)java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
 form2 := HermitianFormFieldReduction

 elif ="" and = "elliptic"and () IsOddIntr)
 and IsOddInt(q) then
 Info(InfoFinInG, 1, "These polar spaces are #over afield L,and consideringL vector K,yields
 alpha := One(f1);
 #map := NaturalEmbeddingByFieldReduction( ps1, f2, alpha, basis );
 =HermitianFormFieldReductionform1,
 java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 elif type1 basis,
 Info(InfoFinInG, 1, "These polar spaces are suitable for
 alpha alpha := One(f1);
 m :=( ps1,, ,basis );
 form2 := BilinearFormFieldReduction(form1,f2,alpha,basis);
 else
 Error("These polar spaces are not suitable for field reduction.");
 fi;
 form1 a form
if type1 =h"and type2 hjava.lang.StringIndexOutOfBoundsException: Range [50, 49) out of bounds for length 55
 Info(InfoFinInG, 1," polar are suitable for field reduction");
 alpha := One(f1);
 #map# <geom1> <>have sameambientjava.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 78
 form2 := QuadraticFormFieldReduction(form1,f2,alpha,basis);
 elif type1 = "elliptic" InstallMethod(NaturalEmbeddingByFieldReduction
 Info(InfoFinInG, 1, "IsProjectiveSpace, IsProjectiveSpace ],
 alpha := One(f1);
 #;
 = (f2alpha,;
 java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 0
 Info(InfoFinInG,java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
=f)
 #map:NaturalEmbeddingByFieldReduction( ,f alpha ;
 :=QuadraticFormFieldReduction(form1f,,)

 # If ps1 is #
hawewilluse for theembedding
# depends theisometry ofps1.Theisometry type is determinedby
 # beingasquareor not wheregammais
 timesthe determinant ofthe bilinear ,dividedby twojava.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76
 # The conditions we use herejava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
 # java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0
#theconditionsin Gill'paper.

 elif type1 = "parabolic" and type2 = "hyperbolic" and #
 Info(InfoFinInG, 1, "java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 0
:java.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 75
 # BilinearFormByQuadraticForm (see documentation of the Forms package).
 gamma:=((-1)^((r-1)/2)) * Determinant(bil1!.matrix)/2;
 if q^(t/2) mod 4 = 1 then # the product of alpha and gamma must be anonsquare
 alpha := First(f1, a -> not IsZero(a) and not#<java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 41
 else # the product of java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 2
 alpha := First(f1,a- () and is_square(a*amma ;
 fi;
 #map := NaturalEmbeddingByFieldReduction( ps1, f2, alpha, basis );
 form2 := QuadraticFormFieldReduction(form1,f2,alpha,basis);

elif =parabolic"and type2 = "elliptic" and IsEvenInt(t) and IsOddInt(q) then
 Info(InfoFinInG, 1, "These polar spaces are suitable for field reduction: Parabolic -> Elliptic");
 bil1:=AssociatedBilinearForm(form1); # important to use this instead of
 # BilinearFormByQuadraticForm (see documentation of the Forms package).
 :(-)(-1))*Determinantbil1.)2;
 if q^(t/2) mod 4
 alpha := First(f1, a -> not IsZero(a) and is_square( a*gamma, f1 ) );
 else # the product of alpha and gamma must be a nonsquare
 alpha := First( [ IsBilinearForm, IsIsBilinearForm,IsField,IsFFE,IsBasis]
 fi;
 # : ( ps1 2 alpha,basis )
 =QuadraticFormFieldReduction(orm1,2,,basis)java.lang.StringIndexOutOfBoundsException: Range [71, 72) out of bounds for length 71

 else
 Error" spacesare ot for reduction.)
 fi;
fi;

c1 := BaseChangeToCanonical( form2 );
 if not IsCanonicalPolarSpace(ps2) then
 if type2 in ["parabolic", "hyperbolic", "elliptic"] then
 c2 := BaseChangeToCanonical(QuadraticForm(ps2));
 else
 c2 := BaseChangeToCanonical(java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 21
 fi;
 cps2 := c2^-1 * c1;
 else
 cps2 := c1;
 fi;
 cps2inv := cps2^-1;

 fun := java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 10
 local mat1,mat2;
 mat1 =subspace.bj;
 java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 5
 mat1f2;
 fi;
 mat2 := BlownUpMat(java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 13
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8

 is (714)

 prefun := function( subspace ) # This map is the inverse of func and returns an error, or a subspace of geom1
 local flag,basvecs,mat1,span,x,v,v1,i,vecs;
 :true;
 if not subspace in ps2 then
 Error("The input is not in the java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 35
 fifunction(il1,f2,alpha)
 if not IsInt((Dimension(subspace)+1)/t) then
 flag:=false;
 else
 basvecs:=BasisVectors(basis);
 vecs := Unpack(UnderlyingObject(subspace))*cps2;
 mat1 := List(vecs,x->List([1..d1],i->x{[(i-1)*t+1..i*t]}*basvecs));
 span := VectorSpaceToElement(AmbientSpace(ps2),BlownUpMat(basis,mat1)*cps2inv); #the ambientspace makes sure that there is no error message here yet.
 if not span = subspace then
 flag := false;
 fi;
 fi;
 if flag = false then
 Error("The input is not in the range of the field reduction map!");
 fi;
 return VectorSpaceToElement(ps1,mat1);
 end;

 #iso := IsomorphismPolarSpacesNC(AmbientGeometry(Range(map)), ps2);
#morphism := GeometryMorphismByFunction(ElementsOfIncidenceStructure(ps1),
 #ElementsOfIncidenceStructure(),
 # x -> iso!.fun(map!.fune );

map := GeometryMorphismByFunction(ElementsOfIncidenceStructure(ps1),
 ElementsOfIncidenceStructure(ps2),
 fun, false, prefun);
true )

 if(asIsCanonicalPolarSpaceps1 ( )or
 HasCollineationGroup( ps1 ) then
 if computeintertwiner then
 hom := function( m )
 imagejava.lang.StringIndexOutOfBoundsException: Range [28, 29) out of bounds for length 28
 image := BlownUpMat(basis, m!.mat);
 ( QuadraticFormFieldReduction
 CollineationOfProjectiveSpace image*cps2inv ;
 end;
 : function(m)
 local preimage;
 preimage := ShrinkMatd1:=(!.atrix;
 ConvertToMatrixRepNC( preimage, f1 );
 return CollineationOfProjectiveSpace(preimage, f1);
 end;
 g1 := IsometryGroup( ps1 );
 :=GeneratorsOfGroup g1);
 newgens := List(gens, hom iin [..]do
 g2 := Group( newgens );
 (, Sizeg))
 od;
 SetIntertwiner( map, twiner);
 fi;
 java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7

(ap );

return map;
endqf2=QuadraticFormByMatrix(f2)

# added 2/7/14 jdb
##################################################
# O NaturalEmbeddingByFieldReduction
# NaturalEmbeddingByFieldReduction polar andreturnsanembedding
#obtainedby field reduction if suchan embedding exists, otherwiseit an .
##
InstallMethod (NaturalEmbeddingByFieldReduction,
"for two classical polar spaces",
[IsClassicalPolarSpace, java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 37
function(ps1, ps2)
return NaturalEmbeddingByFieldReduction(,s2,rue;
 basis=(AsVectorSpace(2,f1!.asefield)java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48

#####################java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
#
# Embeddings by SUBFIELDS
#
#
############################################## either form ahermitian form.

# 11/ jdb
# found another cvec/# extra functionality to ,and isused thirdargument thejava.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
#########################################java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
#O NaturalEmbeddingBySubfield( <geom1>, <geom2> )
# returns the embedding of <geom1> in spaces the
# same
# Install( HermitianFormFieldReduction,
##
InstallMethod( "for a hermitian form and a field
fortwo spaces,
[ function(hf1,2,lphabasis
function( geom1, geom2 )
 local map, f1, f2, func, prefun, g1, g2,
 gens1, gens2, hom, twinerfunc, twinerprefun;
 f1 := geom1!.basefield;
 f2 := geom2!.basefield;
 if geom1!.dimension if geom1!.dimension always a
 " / sizes incompatible";returnjava.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
 fi;

 func := x -> VectorSpaceToElementV2:=f2^d2;

 prefun := function( x )
 local xmat;
 if not ForAll( Flat(x!.obj), i -> i in f1 ) then
 Error("Element does not have all of its coordinates in ", f1);
 fi;
 xmat := Unpack(x!.obj);
 if x!.type = 1 then
 ConvertToVectorRep(xmat);
 else
 ConvertToMatrixRep(xmat);
 fi;
 return VectorSpaceToElement(geom1, xmat ); # the same here.
 end;

 map := GeometryMorphismByFunction( ElementsOfIncidenceStructure(geom1),
 ElementsOfIncidenceStructure(geom2),
 func, false, prefun );
 SetIsInjective( map, true );

 ## Intertwiner...

 twinerfunc := function( x )
 local xmat;
 if not IsOne(x!.frob) then
 Info(InfoFinInG, 1, "<el> is not in the source of the intertwiner");
 return fail;
 fi;
 xmat := x!.mat;
 return CollineationOfProjectiveSpace( Unpack(x!.mat), f2 ); #here was an unpack needed!
 end;

 twinerprefun := function( y )
 local ymat,q;
 if not IsOne(y!.frob) then
 Info(InfoFinInG, 1, "<el> is not in the range of the intertwiner");
 return fail;
 fi;
 ymat := Unpack(y!.mat); #here too!
 q := ConvertToMatrixRep(ymat);
 if not q = Size(f1) then
 Info(InfoFinInG, 1, "<el> is not in the range of the intertwiner");
 return fail;
 fi;
 return CollineationOfProjectiveSpace(ymat,f1);
 end;

 g1 := ProjectivityGroup( geom1 );
 gens1 := GeneratorsOfGroup( g1 );
 gens2 := List(gens1, twinerfunc );
 g2 : ow=[]
 for j [.d]do
 hom := GroupHomomorphismByFunction(g1, g2, twinerfunc, twinerprefun);
 etIntertwinermap hom)java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
 return map;
end );

# CHECKED 25/7/14 jdb. With new (and correct now) prefun.
# cmat changes 21/3/14 and 24/3/2014, after a nice cultural weekend :-)
#############################################################################
#O NaturalEmbeddingBySubfield( <geom1>, <geom2>, <bool> )
# returns the embedding of <geom1> in <geom2>, two polar spaces of the
# same dimension, over the fields K and L respectively, where L is an java.lang.StringIndexOutOfBoundsException: Range [0, 79) out of bounds for length 16
# of K. An Intertwiner is computed if <bool> is true.
# The prefun principle is taken from NaturalDualityHermitian.
##
InstallMethod( NaturalEmbeddingBySubfield,
"for two polar spaces and a boolean",
[ IsClassicalPolarSpace, IsClassicalPolarSpace, IsBool ],
function( geom1, geom2, computeintertwiner )
 local f1,f2,q1,q2,gamma, func, prefun ,g2 gens1,gens2,f
;
ge ,d,n ,one i c1,c2 form2;

 #
 # Check Kleidman and Liebeck Table 4.5.A
 # There, the conditions on the polarity
 #
 possiblecases:
 # W -> W
 >H quadratic)
 # O^eps -> H (q odd,
 # H -> H (odd degree)
 # O^eps -> O^eps' (eps = eps'^r)
 #

 f1 := geom1!.basefield; q1 := Size(f1);
 f2 := geom2!.basefield; q2 IsHermitianForm IsField,IsFFE ],
 type1 := PolarSpaceType(geom1);
 type2 := PolarSpaceType(geom2);
 form2 := SesquilinearForm( geom2 ); #will be changed in last case only
 dimension= geom2!.dimension and(2 f1 then
 r := LogInt(q2, q1);
 gram #here thebasefield is always square
 if [type1, type2] = ["symplectic", "hermitianjava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
 gamma := First( f2, t -> not end ;
 newgram := gamma java.lang.StringIndexOutOfBoundsException: Range [0, 1) out of bounds for length 0
 form := HermitianFormByMatrix(newgram, f2);
 elif [type1, type2] = ["symplectic", "symplectic"] then
 form := BilinearFormByMatrix(gram, f2);
 elif [type1, type2] = ["hermitian", "hermitian"] and IsOddInt(r) then
 form := HermitianFormByMatrix(gram, f2);
 elif type1 in ["elliptic", "parabolic", "hyperbolic"] and
 type2 = "hermitian" and r=2 and IsOddInt(q2) then
 form := HermitianFormByMatrix(gram, java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
 elif (IsOddInt(r) and type2 in ["elliptic", "parabolic", "hyperbolic"] and
 type1 = type2) or (IsEvenInt(r) and [type1, type2] in [["elliptic","hyperbolic"],
 ["hyperbolic","hyperbolic"], ["parabolic","parabolic"]]) then
 gram := GramMatrix( QuadraticForm(geom1) );
 (,f2;
 form2 := QuadraticForm(geom2);
 else
 Error("Not possible for these geometries\n");
 fi;
 else
 Error("Dimensions and/or field sizes are incompatible"); return;
 fi;

 c1 := BaseChangeToCanonical( form );
 if not IsCanonicalPolarSpace( geom2 ) then
 c2 := BaseChangeToCanonical( form2 );
 change := c2^-1 * java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 20
 else
 change : : . +1;
 fi;
 #1.thepolarspaceisoforthogonal type

 d := geom1!.dimension+1;
 basis := Basis(AsVectorSpace(f1,f2));
 n := Length(basis);
 bmat := NullMat(d,d*n,f1);
 one := One(f1);
 for i in [1..d] do
 bmat[i,1+(i-1)*n] := one;
 od;

 func := function( el )
 return VectorSpaceToElement(geom2,Unpack(el!.obj) * invchange);
 end;

 prefun := function( el )
 local qf2:QuadraticFormFieldReduction(qf1,2,,)java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
 vec := Unpack(el!.obj) * change;
 if el!.type = 1 then
 nvec := vec / First(vec,x->not IsZero(x));
 if not ForAll( nvec, i -> i in f1 ) then
 Error("Element is not in the range of the if IsSingularForm(f2) then
 fi;
 ConvertToVectorRepNC(nvec,1; #ecessary alreadyover.
 else
 bgen=BlownUpMat(basis,)
 list : SumIntersectionMat(gen )2;#hard Meetoperation -java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
 ,x>ShrinkVec(2f1,
 if not (IsMatrix(nvec) and Length(nvec) = el!.type) then
 #return fail;
 Error("Element is not in the range of the geometry morphism");
 fi;
 ConvertToMatrixRepNC(nvec,f1);
 fi;
 return =em.java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
 end;

 map := GeometryMorphismByFunctionlocal mat1mat2;
 ElementsOfIncidenceStructure(geom2),
 func, false, prefun );
 SetIsInjective( map, true );

 twinerfunc mat1 := subspace!.obj;
 local xmat;
 if not IsOne(x!.frob) then
 Info(InfoFinInG, 1, "<el> is not in the source of the intertwiner");
 return fail;
 fi;
 xmat := x!.mat;
 return CollineationOfProjectiveSpace( change * Unpack(x!.mat) * invchange, f2 ); #here was an unpack needed!
 end;

 twinerprefun := function( y )
 localmat2 :=BlownUpMatbasis,at1;
 if not IsOne(y!.frob) then
 Info(
 return fail;
 fi;
 invchange*Unpack(!. * change;# too
 : /(mat[1]x> IsZerox));
 (ymat,i i in f1 ) then
 Info(InfoFinInG, 1, "<el> is not in the range of the intertwiner");
 return fail;
 else
 ConvertToMatrixRepNC(ymat,f1);
 return CollineationOfProjectiveSpace(ymat
 fi;
 endjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12

 HasCollineationGroup( geom1
 g1 := SimilarityGroup( geom1 );
 gens1 := GeneratorsOfGroup( g1 );
 gens2 := List(gens1, twinerfunc );
 g2 := Group(gens2);
 SetSize(g2, Size(g1));
 hom := GroupHomomorphismByFunction(g1, g2, twinerfunc, twinerprefun);
 SetIntertwiner(map, hom);
 else
 Info(InfoFinInG, 1, "No intertwiner computed. <geom1> must have a collineation group computed");
 fi;

 return map;else

end );

#CHECKED//11
############################################
span: (AmbientSpace(s2BlownUpMatb,mat1))
# java.lang.StringIndexOutOfBoundsException: Range [36, 9) out of bounds for length 62
##
InstallMethod( java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 15
"for two polar spaces",
[,IsClassicalPolarSpace,
function( geom1, fi
 NaturalEmbeddingBySubfield geom1,, ;
end );

#########################################java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
#
# PROJECTIONS
#
###################################################java.lang.StringIndexOutOfBoundsException: Range [1, 2) out of bounds for length 1

# CHECKED 28/09/11 jdb
#####################################
#O NaturalProjectionBySubspace( <ps>, <v> )
# returns the morphism hominv := function( m )
# subspace <v>. It is checked if <v> is a subspace java.lang.StringIndexOutOfBoundsException: Range [0, 53) out of bounds for length 31
##
InstallMethod( NaturalProjectionBySubspace,
"for a projective space and a subspace of theend;
[ IsProjectiveSpace, IsSubspaceOfProjectiveSpace ],
function(ps, =Listgens,)java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39

 ## map from geometry of := GroupHomomorphismByFunction ,hominv;
 ## to the projective space of lesser dimension

 local psdim, b, vdim, vs, sub, func, pre,
 mb,,gen,basimg
 canbas, zero, fi;
 if not v in ps then
 Error("<v> is not java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 fi;

 f := ps!.basefield;
 b := Unpack(v!.obj); #cmat # version: user a alpha and basis,agreeswith bool=java.lang.StringIndexOutOfBoundsException: Range [80, 81) out of bounds for length 80
 vdim := v!.type;
 if vdim = 1 then b:=[b]; fi;
 vs := ps!.vectorspace;
 sub := Subspace(vs, b);

 ## if v has dimension j, then the new geometry
 ## has dimension psdim - vdim

 ps2 := ProjectiveSpace( psdim - end );

 ## This code was adapted from Thomas Breuer's
 #"aturalHomomorphismBySubspace"code

 Wvectors:= b;
 =MutableBasis( f,Wvectors ;
 : ]java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
 for gen in BasisVectors( Basis( vs ) ) do
 if not IsContainedInSpan( mb, gen ) then
 Add( compl, gen );
 NaturalEmbeddingByFieldReductif2,lphaBasis(AsVectorSpace(2,(ps1),)
 fi;
 od;
 bas := BasisNC( vs, Concatenation( Wvectors, compl ) );

 # Compute the linear mapping by images.
 img:= FullRowModule( f, Length( compl ) );
 canbas:= CanonicalBasis( img );
 zero:= Zero#returns ( <>, <> <alpha,<basis, )
 transformation want
 basimgs:= Concatenation( ListWithIdenticalEntriesInstallMethod(NaturalEmbeddingByFieldReduction,
 canbas ;

 func := function( x )
 y
 if not v in x then
 end)java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
 fi;
 y := List(Unpack(Unwrap(x)),i-> Coefficients(bas,i))*basimgs; #was x^_, now UnWrap(x), cmat unpack
 if not IsEmpty(y) then
 ## Note: TriangulizeMat does not return a matrix of full
 # rank, whereasSemiEchelonMatdoes!
 y := SemiEchelonMat(y)!.vectors; ## JB: 3/11/2012. Moved this line from the "else" to here so it also applies to the first case.

 if x!.type - vdim = 1 then

 ConvertToVectorRepy, f)
 fi;
 #else
 # ConvertToMatrixRepNC(y, f);
 #fi;
 fi;
 if not IsEmpty(y) then
 #return Wrap(ps2, x!.type - vdim, y);
 return VectorSpaceToElement(ps2,y);
 else
 return EmptySubspace(ps2);
 fi;
 end;
p: function(y)
 local x;
 Unpack(!obj; #mat unpack
 y!.type=1then x : [] fi;
 x := Concatenation(x * compl, b);
 #ConvertToMatrixRepNC(x, f);
 return VectorSpaceToElement(ps, x);
 end;
 map ##########################################
 java.lang.StringIndexOutOfBoundsException: Range [67, 66) out of bounds for length 85
 return map;
end );

 28/911
#######################################################################[sClassicalPolarSpace]
#O NaturalProjectionBySubspaceNC( <ps>, <v> )
# is likeits namesake except that it
## has no checks
##
InstallMethod( java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 22
fora java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 50
[ IsProjectiveSpace, IsSubspaceOfProjectiveSpace ],
function(ps, v)

 ## map from geometry of elements incident ## map from geometry of elements incident with
 ## to the projective space of lesser ffunction ,

 local psdim, b, vdim, vs, sub, func, pre,
ors ,compl gen, bas ,
 changed with 2714

 psdim := ps!.dimension;
 f := ps!.basefield;
 =v!.bj
 vdim := v!.type;
 if vdim = 1 then b:=[b]; fi;
 vs := ps!.vectorspace;
 sub := java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 0

 #v dimension j thenthenewgeometry
 ## has dimension psdim - vdim

 - (t odd)

 # codewas adapted from Breuer'
" code

 Wvectors:= b;
 mb:= MutableBasis( f, Wvectors );
 compl:= [];
 for gen in BasisVectors( Basis( vs #Q+ ->Q+(lwaysjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
 if not IsContainedInSpan( mb, gen ) then
 Add( compl, gen );
 CloseMutableBasis( mb, gen );
 fi;
 od;
 bas := BasisNC( local t, r, type1,, ,f1,f2,newps ,map,

#the java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 47
 img:= FullRowModule =PolarSpaceTypeps1)
 canbas:= CanonicalBasis( img );
 zero:= Zero( img );
 # Matrix of the linear transformation we want
 basimgs:= Concatenation( ListWithIdenticalEntries( fi;
 BasisVectors canbas );

 func := function( x )
 local y;
 y :List(!obji (basi)basimgs
 :SemiEchelonMat).ectors
 if x!.type - vdim = 1 then
 y := y[1];
 (,f);
 java.lang.StringIndexOutOfBoundsException: Range [18, 19) out of bounds for length 18
 ConvertToMatrixRepNC(if IsSesquilinearFormform1 then
 fi;
 if not IsEmpty(y) then
 return Wrap(ps2, x!.type - vdim, y);
 else
 return EmptySubspace(ps2);
 fi;
 end;

 pre := function( y )
 local x;
 x := y!.obj;
 if y!.type = 1 then x := [x]; fi;
 x := Concatenation(x * compl, b);
 (,f);
 return VectorSpaceToElement(ps, x);
 end;

 map := GeometryMorphismByFunction(ElementsOfIncidenceStructure(ps),
 ElementsOfIncidenceStructure(ps2), func, pre );
 return map;
 end );

# CHECKED 28/09/11 jdb
###################################
O\(<s> <>)
# returns the quotient space of theform2=java.lang.StringIndexOutOfBoundsException: Range [49, 48) out of bounds for length 71
subspace<ps
#
InstallOtherMethod(\/,
"for a projective space and a subspace",
[ IsProjectiveSpace and IsProjectiveSpaceRep #ap :NaturalEmbeddingByFieldReduction ,basis)
function ps,v)
 if not v in ps then
" not to the projective space" );
fi
returnRange NaturalProjectionBySubspace( v)).;
end );

# CHECKED else
#cmat change 21/3/14
#############################################################################
#O NaturalProjectionBySubspace( <ps>, <v> )
# returns the morphism from the polar space to theelse aquadraticform
# subspace <v>. It is checked if <v> is a subspace of <ps>.
##
InstallMethod( NaturalProjectionBySubspace,
 "for a polar space and a subspace",
 [ elif type1 = "elliptic" and type2 = "elliptic" then
function(ps, Info(InfoFinInG, 1, " suitable for reduction)java.lang.StringIndexOutOfBoundsException: Range [78, 79) out of bounds for length 78

 elements with v
 # thepolars of sametypebut lesser dimension
 local pstype, psdim, b, vdim, vs, func, pre,
 Wvectors, mb, java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 20
 ps2, map, hom, a, m, newform, f, perp;
 if not v in ps then
Error(<>is not an element of <ps>)java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
 fi;
 pstype := PolarSpaceType(ps);
 : !dimension
 =ps.;
 = (v!obj;# unpacknecessary for.
 vdim : and linear setsin geometry,preprint,theyare
 vdim =1 then :[] fi
 perp := PolarMap(ps);
 vs := VectorSpace(f, Unpack(perp(v)!.obj)); #PolarMap maps to elements, containing a cmat.

 ## AssociatedBilinearForm(form1) #important to use this instead of
 ## has dimension psdim - 2*vardim

 Wvectors:= b;
 b= utableBasis( f, Wvectors );
 compl:= [];

 if not IsContainedInSpan( mb alpha=First(f1 - IsZero() is_square( agamma,f1 )
 (,gen );
 CloseMutableBasis( mb, gen );
 fi;
 od;
 bas := BasisNC( vs, Concatenation( Wvectors, compl ) );

 # Compute the linear mappingelif type1="parabolic"andtype2= e" and ()and () then
 :FullRowModulef compl))java.lang.StringIndexOutOfBoundsException: Range [46, 47) out of bounds for length 46
 canbas:= CanonicalBasis( img );
 zero:= Zero( img );
 # Matrix of the linear transformation we want
 , v - zero )
 if q^(t/2) mod 4 = 1 then # the product^t2 mod4= 1 #productofalpha and gamma must asquare
 C(,f;#seles now.
 # product ofalpha be

 if HasQuadraticForm( ps ) then
 m := QuadraticForm(ps)!.matrix;
 newform := QuadraticFormByMatrix(aform2 =QuadraticFormFieldReductionform1,2,lpha,basis;
 else
 m := Error(Thesepolarspaces not forfieldreduction"java.lang.StringIndexOutOfBoundsException: Range [70, 71) out of bounds for length 70
 if pstype = "hermitian" then
 # c1 := BaseChangeT( form2 )
 newform := a^java.lang.StringIndexOutOfBoundsException: Range [83, 82) out of bounds for length 92


 newform := c2 := BaseChangeToCanonical(SesquilinearFormp));
 fi;
 fi;
 ps2


 func := function( x )
 y;
 if not v in x then
 Error("Subspace is notif .
 fi;
 if not x in ps then
 Error" is element of the space")
 fi;
 if=x return
 EmptySubspace(ps2);
 else
 y =List(npack(x!.bj,->Coefficients(as,)*; cmat here
 y := SemiEchelonMat(y)!.vectors;
 if x!.type - vdim = 1 then Error(Theinputnotintherangethe reductionmap";
 y := y[1]; fi
 #ConvertToVectorRep(y, f);
 #else
 # ConvertToMatrixRepNC(y, f);
 java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
 return VectorSpaceToElement(ps2, y);
 java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
 end;

 pre := function( y )
 local x;
 if not y in ps2 then
 Error("Subspace is not an element of the polar space");
 fi;
 x := Unpack(y!.obj); #cmat unpack here.
 if y!.type = 1 then x := [x]; fi;
 x := Concatenation(x * compl, StructuralCopy( b ));
 #ConvertToMatrixRepNC(x, f); #useless now Error"The input is not in the range of the field reduction!";
 return VectorSpaceToElement(ps, x);
 end;
 map := GeometryMorphismByFunction
 java.lang.StringIndexOutOfBoundsException: Range [67, 66) out of bounds for length 92
 return map;
 end );

# CHECKED 28/09/11 jdb
################################################################(s2,
#O NaturalProjectionBySubspaceNC( <ps>, <v> )
# isjustlikeits namesake exceptthat
## has no checks
##
InstallMethod( NaturalProjectionBySubspaceNCHasCollineationGroup ps1 ) then
 "for a polar space and a java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 32
[java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 66
 function(ps, v)

 ## map from geometry of elements incident with v
 ## to the polar space of the same type but of lesser dimension =(,*m. ;

 localpstype, psdim,b,vdim, vs, func,pre
 Wvectors, mb, compl, gen, bas, img, canbas, zero, basimgs,
 gens: ( ;

 :=(ps;
 psdim := ps!.dimension;
 f := ps!.basefield;
 b := v!.obj;
 vdim := v!.type;
 if vdim = 1 then b
 perp := PolarMap(ps);
 s : VectorSpace(f,perp(v).;

 # java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 51
 ## has dimension psdim - 2*vdim

 Wvectors:= b;
 mb:= MutableBasis( f, Wvectors );
 compl:= [];
 forreturn (,s2);
 if not IsContainedInSpan( mb, gen ) then
 Add( compl, gen );
 CloseMutableBasis(###################################
 fi;
 od;
 bas := BasisNC( vs, Concatenation( Wvectors, compl ) );

 # Compute the linear mapping by images.
 img:= FullRowModule( f, Length( compl ) );
 canbas:= CanonicalBasis( img );
 zero:= Zero( img );
 # Matrix of the linear transformation we want
 basimgs:= Concatenation( List( Wvectors, v -> zero ),
 BasisVectors( canbas ) );
 ConvertToMatrixRep(basimgs, f);
 a := compl;

 if HasQuadraticForm( ps ) then
 m := QuadraticForm(ps)!.matrix;
 newform := QuadraticFormByMatrix(a * m * TransposedMat(a), f);
 else
 m := java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 1
 if pstype = "hermitian" then
 newform := HermitianFormByMatrix(a * m * TransposedMat(a^CompanionAutomorphism(ps)), f);
 else
 newform := BilinearFormByMatrix(a * m * TransposedMat(a), f);
 fi;
 fi;
 ps2 := PolarSpace( newform );

 func := function( x )
 local y;
 y := List(x!.obj,i-> Coefficients(bas,i))*basimgs;
 y := := SemiEchelonMaty!v;
 x. -vdim =1 then
 y := y[1];
 ConvertToVectorRepNC(y, f);
 else
 ConvertToMatrixRepNC(y, f);
 fi;
 ( <> <>
 end;

 pre := function( y )
 local x;
 x := y!.obj;
 if y!.type = 1 then x := [x]; fi;
 x := Concatenation( ,
 ConvertToMatrixRepNC(x, f);
 return VectorSpaceToElement(ps, x);
 end;
 map=GeometryMorphismByFunction(lementsOfIncidenceStructure)
 ElementsOfIncidenceStructure(ps2), func, false, pre );
 return map;
 end );

#############################################################################
#
# Klein correspondence
#
#############################################################################

#############################################################################
# two helper operations. They can be used by the users, but on the other In directoryare the examples all removed.For
# the form of the Klein quadric is fixed, so this is against the philiosphy of
# FinInG, at least for user functions. Also, there are no checks. So we will
# describe these in the technical section of the documentation.
#############################################################################

# CHECKED 28/09/11 jdb
# changed 28/3/14 jdb
#############################################################################
#O PluckerCoordinates( <lobj> )
# returns the Plucker coordinates of the line represented by <lobj>. We accept
# a matrix representing the line. No check on whether this is a line of PG(3,q),`
# so use with care. cmat note: whether l is a cmat or not, coords will be a plain
# list (so no cvec). This is the internal version accepting a matrix.
##
InstallMethod( PluckerCoordinates,
"for a matrix representing a line of PG(3,q)",
 [ IsMatrix ],
function( lobj )
 local pij, u, v, coords;
 pij := function(u,v,i,j)
 return u[i]*v[j] - u[j] * v[i];
 end;
 u := lobj[1];
 v := lobj[2];
 coords := [pij(u,v,1,2),pij(u,v,1,3),pij(u,v,1,4),
 pij(u,v,2,3),pij(u,v,4,2),pij(u,v,3,4)];
 return coords;
end );

# CHECKED 28/09/11 jdb
# changed 28/3/14 jdb
#############################################################################
#O PluckerCoordinates( <line> )
# returns the Plucker coordinates of the line represented by <lobj>. We accept
# a matrix representing the line. No check on whether this is a line of PG(3,q),`
# so use with care. cmat note: whether l is a cmat or not, coords will be a plain
# list (so no cvec). This is the user version acceptingxmat : Unpack(!obj;
# space.
##
#InstallMethod( PluckerCoordinates,
# "for a matrix representing else
# [ IsSubspaceOfProjectiveSpace ],
# VectorSpaceToElement,xmat) #t same herejava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
# localmap =GeometryMorphismByFunction(geom1,
# !type< 2 (!geo)< java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
# Error(" <l> must be a line of a PG(3,q)");
# fi
# lobj := l!.obj;
# pij := function(u,v,i,j)
# return u[i xmat;
# end;
# u := lobj[1];
# v := lobj[2];
# coords := [pij(u,v,1,2),pij(fi
# pij(u,v,2,3),pij(u,v,4,2),pij(u,v,3,4)];
# return coords;
# java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0

0911jdb
# changed 28/3/14 jdb
###################################ymat : (!mat;#ere !
# InversePluckerCoordinatesvar )
# returnsalist twovectors the with Pluckercoordinates var>
# no check on whether this is really a line of the particular Q Info(InfoFinInG,1 < isnotintherangeofthe intertwiner)java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
# caution: the list l is not Triangulized!
# .
##
InstallMethod( InversePluckerCoordinates,
"for a vector repr. a point of Q+(5,q): X0X5+X1X4+X2X3=0",
 [IsVector]

 ## The point must satisfy the Pluecker relation x1x6+x2x5+x3x4=0.
function( x )
 returnmap
 pairs := [[1,2],[1,3],[1,4],[2,3],[4,2],[3,4]];
 i := PositionNonZero(x);
 pair := pairs[i];
 zero := Zero(x[1]);
 l := [];
 if 1 in pair then
 Add(l, [ zero, x[1], x[2], x[3] ]);
 fi;
 if 2 in pair then
 Add(l, [ x[1], zero, -x[4], x[5] ]);
 fi;
 if 3 in pair then
 Add(l, [ x[2], x[4], zero, -x[6] ]);
 fi;
 if 4 in pair then
 Add(l, [ x[3], -x[5], x[6], zero ]);
 fi;
 return l;
end );

#############################################################################
# Klein correspondence: the user operations.
#############################################################################

# added may 2014 jdb
#############################################################################
#O PluckerCoordinates( <l> )
# returns the Plucker coordinates of the line <l>. The is the user variant
# of the operation. It accepts a line of PG(3,q), does some checking, and
# uses the helper operation.
##
InstallMethod( PluckerCoordinates,
"for a line of PG(3,q)",
 [ IsSubspaceOfProjectiveSpace ],
function( l )
 if Dimension(AmbientSpace(l)) <> 3 then
 Error("<l> is not a line of a three dimensional projective space");
 elif l!.type <> 2 then
 Error("<l> is not a line");
 fi;
 return PluckerCoordinates(l!.obj);
end );

# added may 2014 jdb
#############################################################################
#O KleinCorrespondence( <f>, <computeintertwiner> )
# returns the well known morphism from the lines of PG(3,f), f a finite field
# to the points of Q+(5,q): x0x5+x1x4+x2x3 = 0. This is the bare essential
# of some geometry morphisms that follow. For didactical reasons, it looks
# useful to me to have this version that maps to a fixed hyperbolic quadric.
# Of course a flexible verion of this operation is also present.
##
# The interwiner: - the frobenius automorphism commutes with taking PluckerCoordinates
# so we only have to deal with the matrices in the twiner/pretwiner
# funcs.
# - A projectivity is known if its image of a frame is known. So:
# going from PGL(4,q) -> PGO+(6,q): the standard basepoints of PG(5,q)
# are points of the quadric, compute the lines (java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 58
# , computer the point (Plucker), this gives the matrix of the element
# of PGO+5,)whereeach is determinedup to factor It turnsout
# using order as in twinerfunc,theneeded factors are alwayst same:
#[6] line.
#same dimension,overthefields L, where is extension
# greekand latinplanesdonot a ofPG3q!
# The principle is the same as above, just more work is needed to compute
# the factors here (they seem not to be independant of the java.lang.StringIndexOutOfBoundsException: Range [0, 68) out of bounds for length 2
k/areswappedisdonebymappingapoint
# to a generator, computing the image, mapping back, and see whether
# the three lines span PG(3,q). If so, we have geom1 , computeintertwiner)
# imageof thepointisaplaneandthe projectivity
# the greeks/atinsand inducesinfactacorrelation PG3q.
#
InstallMethod( KleinCorrespondence,
"for a finite field and a boolean",
 [ IsField, IsBool ],
function( f, computeintertwiner )
 local i, form, map, pg, mat, pre, #All possiblecasesjava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
 twinerprefun, coll1, gens1, coll2, gens2, hom, id;

 :=O()

 ## The form for the resulting f1 := geom1!.basefield; q1 := Size(f1);
 ## x1x6+x2x5+x3x4 = 0

 mat := NullMat(6, 6, f);
 for i in [1..3] do
 mat[i,7-i] := one;
 od;
 form : iftype1, type2 "" "ermitian ]and r then
:PolarSpace form ;
 :=ProjectiveSpace(,)java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29

 :
 l)
 local pt1;
 pt1 := PluckerCoordinates( l!.obj );
 return VectorSpaceToElement(ps, pt1);
 end;

 inv :=
 function( x )
 local l;
 l := InversePluckerCoordinates [hyperbolic,hyperbolic] ",parabolic"])then
 VectorSpaceToElement,l)
java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 15

 map := GeometryMorphismByFunction(Lines(pg) Error(Notpossible forthesegeometriesn)java.lang.StringIndexOutOfBoundsException: Range [49, 50) out of bounds for length 49
 SetIsBijective(map, true);

 twinerfunc := function(g)
 c1: ( form)
 mat := Unpack(g!.mat);
 newmat := [];
 newmat[1] := PluckerCoordinates
 newmat[2] := PluckerCoordinates([mat[3],mat[1]]);
 newmat[3] := java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 25
rdinates[3,]java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
 newmat[5] =;
 newmat[6] := -PluckerCoordinates([-mat[4],mat[3]]);
 return ProjElWithFrob(newmat,g!.frob,f);
 end;

 id := IdentityMat(4, f);

 twinerprefun := function( g )
 local mat, newmat, lines, pts, ipts, ilines, e, x, ept,
 ielines, ie, cs, iept;
 mat := Unpack(g!.mat);
 lines:= [];
 lines[1] := [[id[1],id[2]],[id[1],id[3]],[id[1],id[4]]];
 lines[2] := [[id[2],id[3]],[id[2],id[4]]];
 lines[3] := [[id[3],id[4]],[id[3],id[1]]];
 lines[4] := [[id[4],id[1]],[id[4],id[2]]];
 pts := List(lines,x->List(x,y->PluckerCoordinates(y)));
 ipts := List(pts,x->x*mat);
 ilines := List(ipts,x->List(x,y->InversePluckerCoordinates(y)));
 if Rank(Union(ilines[1])) = 3 then
 Info(InfoFinInG, 1, "<el> is not inducing a collineation of PG(3,q)");
 return fail;
 fi;
 ipts := List(ilines, x->SumIntersectionMat(x[1],x[2])[2]);
 newmat := List(ipts,x->x[1]);
 e := [1,1,1,1]*one;
 ept : List([,id[1],e,d2],->luckerCoordinatesy))java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
 iept := List(ept,x->x*java.lang.StringIndexOutOfBoundsException: Range [0, 28) out of bounds for length 20
 ielines := List(iept,x->InversePluckerCoordinates(x));
 ie := SumIntersectionMat(ielines[1],ielines[2])[2];
 cs := SolutionMat(newmat,ie[1]);
 for i in [1..4] do
 newmat[i] := newmat[i]*cs[i];
 od;
 return ProjElWithFrob(newmat,g!.frob,f);
 end;

 if computeintertwiner then
 coll1 := CollineationGroup(PG(3,f));
 gens1
 gens2 := List(gens1, twinerfunc);
 coll2 := GroupWithGenerators(gens2);
 hom := GroupHomomorphismByFunction(coll1, coll2, twinerfunc, twinerprefun);
 SetIntertwiner( map, hom );
 fi;

 return map;
end );

#############################################################################
#O KleinCorrespondence( <q> )
# returns KleinCorrespondence( GF(q), true )
##
InstallMethod( KleinCorrespondence,
"for a prime power",
 [ IsPosInt ],
q -> KleinCorrespondence(GF(q),true)
);

#############################################################################
#O KleinCorrespondence( <q> )
 Error("lement is not in rangeof the java.lang.StringIndexOutOfBoundsException: Range [56, 55) out of bounds for length 67
##
InstallMethod( KleinCorrespondence,
"for a prime power and a boolean",
 [ IsPosInt, IsBool ],
function(q, computeintertwiner )
 return KleinCorrespondence(GF(q),computeintertwiner);
end );

#############################################################################
#O KleinCorrespondence( <f> )
# returns KleinCorrespondence( GF(q), true )
##
InstallMethod( KleinCorrespondence,
"for a finite field",
 [ IsField ],
f -> KleinCorrespondence(f,true)
);

# CHECKED 28/09/11 jdb
#############################################################################
#O KleinCorrespondence( <quadric>, <computeintertwiner> )
# returns the well known morphism from the lines of PG(3,q) to the points
# of Q+(5,q). Of course, the bilinear form determining the latter, can be chosen
# by the users. This method is exactly the same as the method for
# KleinCorrespondence( <f>, <computeintertwiner> ), but adds base changes.
##
InstallMethod( KleinCorrespondence,
"for a hyperbolic quadric and a boolean",
[ IsClassicalPolarSpace, IsBool ],
function( quadric, computeintertwiner )
 local f, i, form, map, pg, mat, twinerfunc, twinerprefun,
 pre, func, ps, iso, one, c, c1, c2, id, cinv, coll1, gens1,
 coll2, gens2, hom;
 if ProjectiveDimension(quadric) <> 5 then
 Error("<ps> is not Klein's quadric");
 fi;
 if not IsHyperbolicQuadric(quadric) then
 Error("<ps> is not Klein's quadric");
 fi;
 f := quadric!.basefield;
 one := One(f);

 ## The form for the resulting Plucker coordinates is
 ## x1x6+x2x5+x3x4 = 0

 mat := NullMat(6, 6, f);
 for i in [1..3] do
 mat[i,7-i] := one;
 od;
 form := QuadraticFormByMatrix(mat, f);
 ps := PolarSpace( form );
 c1 := BaseChangeToCanonical( form );
 if not IsCanonicalPolarSpace(quadric) then
 c2 := BaseChangeToCanonical(QuadraticForm(quadric));
 c := c2^-1 * c1;
 else
 c := c1;
 fi;
 cinv := c^-1;
 pg := ProjectiveSpace(3,f);

 func := function( el )
 local pt1;
 if el!.type <> 2 then
 Error("<el> is not a line ");
 fi;
 pt1 := PluckerCoordinates( el!.obj );
 return VectorSpaceToElement(quadric, pt1 * cinv ); #base change is here
 end;

 pre := function( el )
 local l;
 if el!.type <> 1 then
 Error("<el> is not a point ");
 fi;
 l := InversePluckerCoordinates( el!.obj * c ); #base change is here
 return VectorSpaceToElement(pg, l);
 end;

 map := GeometryMorphismByFunction(Lines(pg), Points(quadric), func, pre);
 SetIsBijective(map, true);

 twinerfunc := function(g)
 local mat,newmat,frob;
 mat := Unpack(g!.mat);
 frob := g!.frob;
 newmat := [];
 newmat[1] := PluckerCoordinates([mat[2],mat[1]]);
 newmat[2] := PluckerCoordinates([mat[3],mat[1]]);
 newmat[3] := PluckerCoordinates([mat[4],mat[1]]);
 newmat[4] := PluckerCoordinates([mat[3],mat[2]]);
 newmat[5] := PluckerCoordinates([mat[4],-mat[2]]);
 newmat[6] := -PluckerCoordinates([-mat[4],mat[3]]);
 return ProjElWithFrob(c * newmat * cinvfi;
 end;

 id := IdentityMat(4, f);

 twinerprefun not ForAll (ymat,iiinf1 )then
 local mat, newmat, lines, pts, ipts, ilines, e, x, ept,
ielines ,cs, iept,frob;
 frob := g!.frob;
 mat := cinv * Unpack(g!.mat) * c^frob; #base change is here.
 lines:= [];
 lines[1] := [[id[1end;
 lines[2] := [[id[2],id[3]],[id[2],id[4]] HasCollineationGroup geom1 )
 lines[3] := [[id[3],id[4]],[id[3],id[1] gens1 : GeneratorsOfGroup(g1)
 lines[4] := [[id[4],id[1]],[id[4],id[2]]] SetSizeg2,Sizeg1)java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
 pts := List(lines,x->List(x,y->PluckerCoordinates(y)));
 : List(tsx-x*);
 ilines := List(ipts,x->List(x,y->InversePluckerCoordinates(y)));
 if Rank(Union(ilines[1])) = 3 then
 Info(InfoFinInG, 1, "<el> is not inducing a collineation of PG(3,q)");
 return fail;
 java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
 ipts := List(ilines, x->SumIntersectionMat(x[1],x[2])[2]);
 newmat := List(ipts,x->x[1]);
 e := [1,1,1,1]*one;
 ept := List([[e,id[1]],[e,id[2]]],y->PluckerCoordinates(y));
 iept := List(ept,x->x*mat);
 ielines : List(,->(x));
 ie := SumIntersectionMat(ielines[1],ielines[2])[2];
 cs := SolutionMat(newmat,ie IsClassicalPolarSpace ]
 for i in [1..4] do
 newmat[i] := newmat[i]*cs[i];
 od;
 return ProjElWithFrob(newmat,g!.frob,f);
 end;

 if computeintertwiner then
 coll1 := CollineationGroup(PG(3,f));
 gens1 := GeneratorsOfGroup(coll1);
 gens2 := List(gens1, twinerfunc);
 coll2 := GroupWithGenerators(gens2);
 hom := GroupHomomorphismByFunction(coll1, coll2, twinerfunc, twinerprefun);
 SetIntertwiner( map, hom );
 fi;

 return map;

end );

#############################################################################
#O KleinCorrespondence( <quadric> )
# returns KleinCorrespondence( <quadric>, true )
##
InstallMethod( KleinCorrespondence,
"for a hyperbolic quadric",
 [ IsClassicalPolarSpace ],
quadric -> KleinCorrespondence(quadric, true)
);

# added may 2014 jdb.
#############################################################################
#O KleinCorrespondenceExtended( <f>, <computeintertwiner> )
# I had really a lot of fun with the above methods and the mathematics behind.
# Therefore I decided to include an "extension" which also computes the image
# of points and planes of PG(3,q) under the Klein correspondence. This morphism
# becomes an isomorphism from PG(3,q) as incidence geometry to the incidence
# geometry consisting of the points and the two systems of generators of Q+(5,q).
# returnsthe morphism from the spaceto thequotient
# KleinCorrespondences.
# When I will have more time, I'll maybe implement the interwiner from the
# CorrelationCollineationGroup(PG(3,q)) to PGO+(6,q).
##
InstallMethod( KleinCorrespondenceExtended,
"for a field and a boolean",
 [ IsField, IsBool ],
function( f, computeintertwiner )#
 local i, form, map, pg, mat, pre, func, ps, one, twinerfunc,
ens1,coll2,gens2,hom em;

 one := One(f);

he form the resulting is
 ## x1x6+x2x5+x3x4 = 0

=(,,f)
fori [.3
 i7i =one;
 od;
 form := QuadraticFormByMatrix(mat, f);
 ps := PolarSpace( form );
 pg := ProjectiveSpace(3,f);

 func := function( el )
 local t, vec, newvec, ptvec, elvec;
 t := el!.type;
 elvec := Unpack(el!.obj);
 if t = 1 then
 vec := BasisVectors(Basis(Lines(el)!.
1],PluckerCoordinatese,vec2]),PluckerCoordinates[lvecvec[3]);
 return VectorSpaceToElement(ps,newvec);
 elif t = 2 then

 elif t = 3 then
 newvec := [PluckerCoordinates( elvec{[1,2]}), PluckerCoordinates( elvec{[2,3]} ), PluckerCoordinates:
 return VectorSpaceToElement(ps,newvec);
 fi;
 end;

 pre := function( el )
 local elvec,t,newvec,lines,space;
 t : el!type;
 elvec := Unpack(el!.obj);
 if t = 1 then
 newvec := ix ofthe we
 return VectorSpaceToElementpg,newvec;
 elif t = 3 then
 newvec := List(elvec,x->InversePluckerCoordinates
 lines := List(newvec,x->VectorSpaceToElement(pg,x));
e: Meet(ines;
 ifIsEmptySubspacespace) then
 return Span(lines);
 else
 space;
 fi;
 elif t = 2 then
 Error("Lines of Klein quadric have no preimage");
 fi;
 end;

 map := GeometryMorphismByFunction(ElementsOfIncidenceStructure(pg), ElementsOfIncidenceStructure(ps), func, pre);
 SetIsBijective(map, true);
 fi;
 if computeintertwiner then
 em := ConvertToMatrixRepNC f;
 hom := Intertwiner(em);
 SetIntertwiner( map, hom );
 fi; #return Wrap(ps2,!.type -,y)

 return map;
end );

########################################
#O KleinCorrespondenceExtended( <q> )
# returns KleinCorrespondenceExtended( GF(q), true )
##
InstallMethod( KleinCorrespondenceExtended,
"for a prime power",
 [ end;
 - KleinCorrespondenceExtended(F()true)
);

# added may 2014 jdb.
############################################
#O KleinCorrespondenceExtended( <quadric>, <computeintertwiner> )
# The same as KleinCorrespondenceExtended, but accepting a user defined
# hyperbolic quadric
##
InstallMethod( KleinCorrespondenceExtended,
"orafield "
 [ IsClassicalPolarSpace, IsBool ],
 function( quadric, computeintertwiner )
 local
 twinerprefun, coll1, gens1, coll2, gens2, hom, em, c, c1, c2, cinv;

 flocal psdim,,vdim,vs,sub,func,pre,
 one := One(f);

 ## The form for the resulting Plucker coordinates is
 #x1x6x2x5+3 =0

 mat := NullMat(6, 6, f);
 v: !vectorspacejava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
 mat[i,7-i] := one;
 od;
 form := java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 0
 ps := PolarSpace( form );
 pg :=ProjectiveSpace(3,f)

 c1 := BaseChangeToCanonical( form );
 if not IsCanonicalPolarSpace(quadric) then
 c2 := BaseChangeToCanonical(QuadraticForm(quadric));
 c := c2^-1 * c1;
 else
 c := c1;
 fi;
 cinv := c^-1;

 func := function( el )
=[]
 t := el!.type;
 elvec := Unpack(el!.obj);
 if t = 1 then
 vec := BasisVectors(Basis(Lines(el)!.java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 11
 : BasisNC(,Concatenation ,compl ))
 return VectorSpaceToElement(quadric,newvec * cinv); #base change is here...
 elif t = 2 img: FullRowModule( f,Length(compl) )
 return VectorSpaceToElement(quadric,PluckerCoordinates(elvec) * cinv); # ... and here ...
 elif t = 3 then
 newvec := [PluckerCoordinates( elvec{[1,2]}), PluckerCoordinates( elvec{[2,3]} ), PluckerCoordinates( elvec{[1,3]} )];
 return VectorSpaceToElement(quadric,newvec * cinv); #... and here.
 fi;
 end;

 pre := function( el )
 local elvec,t,newvec,lines,space;
 t := el!.type;
 elvec := Unpack(el!.obj) * c; # base change is here.
 if t = 1 then
 newvec := InversePluckerCoordinates( elvec );
 return VectorSpaceToElement(pg, newvec);
 elif t = 3 then
 newvec := List return (s2 !type-vdim ;
 lines := List(newvec,x->VectorSpaceToElement(pg,x));
 space := Meet(lines);
 if IsEmptySubspace(space) then
 return Span(lines);
 else
 return space;
 fi;
 elif t = 2 then
 Error("Lines of Klein's quadric have no preimage in the source of ");
 fi;
 end;

 map := GeometryMorphismByFunction(ElementsOfIncidenceStructure(pg), ElementsOfIncidenceStructure(quadric), func, pre);
 SetIsBijective(map, true);

 if computeintertwiner then
 em := KleinCorrespondence(quadric,true);
 hom := Intertwiner(em);
 SetIntertwiner( map, hom );
 fi;

 return map;
java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

######return ;
#O KleinCorrespondenceExtended( <quadric> )
# returns java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 22
#
InstallMethod( KleinCorrespondenceExtended,
"for a prime power",
 [ IsClassicalPolarSpace ],
quadric -> KleinCorrespondenceExtended(quadric,true)
);

####################I(\/,
#
# DUALITIES between W(q) and Q(4,q) and H(3,q^2) and Q-(5,q)
#
#############################################################################Error "ubspacedoesnot to space")

#############################################################################
# two helper operations. These were derived from the previous version of
# NaturalDuality. The new NaturalDuality will java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
#############################################################################

# Added april 2014. jdb
#############################################################################
#O NaturalDualitySymplectic( <w>, <q4q>, <computeintertwiner>, <reverse> )
# returns the well known isomorphism between W(3,q) and Q(4,q).
# Both polar spaces may be user defined.
# Setup: - Plucker Coordinates map lines of PG(3,q) on points of Q+(5,q): X0X5+X1X4+X2X3 = 0
# - restricting to lines of W(3,q), the range is the points of Q(4, fi;
# intersecting with the hyperplane X0+X5 = 0.
# - When using PluckerCoordinates, we only need the first 5 coordinates. When using java.lang.StringIndexOutOfBoundsException: Index 104 out of bounds for length 6
# makethe sixthcoordinate equal to minusthe first.
# - points of W(3,q): take its pole under the polarity associated with W(3,q), this defines a plane.
# From this plane's underlying object, we have three lines -> three points of Q+(5,q) spanning a generator
# then intersect this generator with X0+X5=0 to find the line of Q(4,q).
# - lines of Q(4,em:=KleinCorrespondence(fcomputeintertwiner;
# we are looking for.
# - some base changes are necessary.
# - all the linear algebra is hard coded.
# - the intertwiners: inspired by the intertwiners of the Klein Correspondence.
# - except for the filters, there is not real check whether the input the correct GQ.
#############################################################################
##
InstallMethod( NaturalDualitySymplectic,
"for a symplectic GQ and a parabolic quadric",
[ IsClassicalGQ, IsClassicalGQ, IsBool, IsBool ],
function( w, q4q, computeintertwiner, reverse )
 local f, one, mat, form_quadric, quadric, data, form_w, cq4q, cw, can,
 func, pre, formw, c1, c2, delta, hyp, coll1, coll2, gens1, gens2, hom,
 twinerfunc, twinerprefun, cq4qinv, cwinv, id;
 f := w!.basefield;
 one := One(f);
 mat := NullMat(5, 5, f);
 mat[1,1] := end )
 mat[2,5] := -one;
 mat[3,4 =one
 java.lang.StringIndexOutOfBoundsException: Range [38, 16) out of bounds for length 50
c1 := BaseChangeToCanonical( form_quadric );
 if not IsCanonicalPolarSpace(q4q) then
 c2 := BaseChangeToCanonical(QuadraticForm(q4q));
 cq4q := c2^-1 * c1;
 else
 cq4q := c1;
 fi;
 cq4qinv := cq4q^-1;
formw := SesquilinearForm( w );
 delta := PolarityOfProjectiveSpace(w);
 hyp := IdentityMat(6,f){[1..5]}; #this and next are in fact HyperplaneByDualCoorindates([1,0,0,0,0,1]..)
 hyp[1,6] := -one;
 if IsCanonicalPolarSpace
 func := function( el )
 ,vec;
 if java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 return VectorSpaceToElement(q4q, java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 16
 else
 plane := el^delta;
 vec := Unpack(plane form : QuadraticFormByMatrixmatf);
 list := [ PluckerCoordinates(vec{[1,2]}), PluckerCoordinates(vec{[2,3]}), PluckerCoordinates(vec{[1,3]}) c : ^1 java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
 plane := SumIntersectionMat(list, hyp)[2]; #hard coded Meet operation :-)
 list := List(plane cinv = ^-;
 return VectorSpaceToElement(q4q, list);
 fi;
 end;
 pre := function( el )
 local vec;
 if el!.type = 1 then #dealing with a point
 vec := Unpack(el!.obj)*cq4q;
 vec[6] := -vec[1];
 return VectorSpaceToElement(w, InversePluckerCoordinates(vec) );
 else
 vec := Unpack(el!.obj)*cq4q;
 vec[1][6] := -vec[1][1];
 vec[2][6] := -vec[2][1];
 return VectorSpaceToElement(w, SumIntersectionMat(InversePluckerCoordinates(vec[1]), InversePluckerCoordinates(vec[2]))[2] );
 fi;
 end;
 else
 cw := BaseChangeToCanonical( formw );
 vec := BasisVectorsBasis(().);
 local list,plane,vec;
 if el!.type = 2 then #dealing with a line
 return VectorSpaceToElement(q4q, PluckerCoordinates(Unpack(el!.obj)*cw^-1){[1..5]} * cq4q^-1);
 else
 plane := el^delta;
 vec := Unpack(plane!.obj)*cw^-1;
 list := [ PluckerCoordinates(vec{[1,2]}), PluckerCoordinates(vec{[2,3]}), PluckerCoordinates( return VectorSpaceToElement(,(elvec *cinv;#..andhere..java.lang.StringIndexOutOfBoundsException: Index 93 out of bounds for length 93
 plane := SumIntersectionMat newvec :[PluckerCoordinates(elvec{12]) PluckerCoordinates(elvec{2,]}) PluckerCoordinates [,]});
 list := List(plane,x->x{[1..5]}*cq4q fi;
 VectorSpaceToElement(,list)
 fi;
 end;
 pre := function( el )
 local vec;
 if el!.type = 1 then #dealing with a point
 vec := Unpack(el!.obj)*cq4q;
 vec[6] := -vec[1];
 return VectorSpaceToElement(w, InversePluckerCoordinates return VectorSpaceToElement(,newvec;
 else;
 vec := Unpack(el!.obj)*cq4q;
 vec[1][6] := -vec[1][1];
 vec[2][6] := -vec[2][1];
 bspaces)then
 VectorSpaceToElement(w, InversePluckerCoordinates(vec[2]) * cw ) );
 fi;
 end;
 fi;

if not reverse then
 data := rec( func := func, pre := pre);
else
 = ( :=, :
fi;

 if computeintertwiner then

 =IdentityMat4,f;

 if IsCanonicalPolarSpace( w ) then

 twinerfunc := function(g)
 local mat,newmat,frob;
 mat := Unpack(#####################################################
 frob := g!.frob;
 newmat := [];
 newmat[2] := PluckerCoordinates([mat[3],mat[1]]){[1..5]};
 newmat[3] := PluckerCoordinates([mat[4],mat[1]]){[1..5]};
 newmat[4] := -PluckerCoordinates([-mat[3],mat[2]]){[1..5]};
 newmat[5] := PluckerCoordinates([mat[4],-mat[2]]){[1..5]};
 newmat[1] := InstallMethod( KleinCorrespondenceExtended,
 return ProjElWithFrob(cq4q * newmat * cq4qinv^(frob^-1),frob,f); #base change is java.lang.StringIndexOutOfBoundsException: Range [0, 89) out of bounds for length 21
 return newmat;
 end;

 twinerprefun := function( g )
 local mat, newmat, lines, pts, ipts, ilines, e, x, ept,
 ielines, ie, cs, iept, frob, i, j;
 : !.frobjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
 mat := cq4qinv * Unpack(g!.mat) * cq4q^############################################
 lines:= [];
 lines[1] := [[id[1],id[3]],[id[1],id[4]]];
 lines[2] := [[id[2],id[3]],[id[2],id[4]]];
 lines[3] := [[id[3],id[1]],[id[3],id[2]]];
 lines[4] := [[id[4],id[1]],[id[4],id[2]]];
 pts := List(lines,x->List(x,y->PluckerCoordinates(y){[1..5]}));
 ipts := List(pts,x->x*mat);
 for i in [1..Length(ipts)] do
 for j in [1..2] do
 ipts[i][j][6] := -ipts[i][j][1];
 od;
 od;
 ilines := List(ipts,x->List(x,y->InversePluckerCoordinates(y)));
 ipts := List(ilines, x->SumIntersectionMat(x[1],x[2])[2]);
 newmat := List(ipts,x->x[1]);
 e := [1,1,1,1]*one;
 ept := List([[e,id[1]+id[4]],[e,id[1]+id[2]]],y->PluckerCoordinates(y){[1..5]});
 iept := List(ept,x->x*mat);
 iept[1][6] := -iept[1][1];
 iept[2][6] := -iept[2][1];

 ielines := List(iept,x->InversePluckerCoordinates(x));
 ie := SumIntersectionMat(ielines[1],ielines[2])[2];
 cs := SolutionMat(newmat,ie[1]);
 for i in [1..4] do
 newmat[i] := newmat[i]*cs[i];
 od;
 return ProjElWithFrob(newmat,g!.frob,f);
 end;

 else

 cw := BaseChangeToCanonical( SesquilinearForm(w) );
 cwinv := cw^-1;

 twinerfunc := function(g)
 local mat,newmat,frob;
 frob := g!.frob;
 mat := cw * Unpack(g!.mat) * cwinv^(frob^-1);
 newmat := [];
 newmat[2] := PluckerCoordinates([mat[3],mat[1]]){[1..5]};
 newmat[3] := PluckerCoordinates([mat[4],mat[1]]){[1..5]};
 newmat[4] := -PluckerCoordinates([-mat[3],mat[2]]){[1..5]};
 newmat[5] := PluckerCoordinates([mat[4],-mat[2]]){[1..5]};
 newmat[1] := PluckerCoordinates([mat[2]+mat[3],mat[1]+mat[4]]){[1..5]} - newmat[2] - newmat[5];
 return ProjElWithFrob(cq4q * newmat * cq4qinv^(frob^-1),frob,f); #base change is here.
 #return newmat;
 end;

 twinerprefun := function( g )
 local mat, newmat, lines, pts, ipts, ilines, e, x, ept,
 ielines, ie, cs, iept, frob, i, j;
 frob := g!.frob;
 mat := cq4qinv * Unpack(g!.mat) * cq4q^(frob^-1);
 lines:= [];
 lines[1] := [[id[1],id[3]],[id[1],id[4]]];
 lines[2] := [[id[2],id[3]],[id[2],id[4]]];
 lines[3] := [[id[3],id[1]],[id[3],id[2]]];
 lines[4] := [[id[4],id[1]],[id[4],id################################################
 pts := List(lines,x->List(x,y->PluckerCoordinates(y){[1..5]}));
 ipts := List(pts,x->x*mat);
 for i in [1..Length(ipts)] do
 for j in [1..2] do
 ipts[i][j][6] := -ipts[i][j][1];
 od;
 od;
 ilines := List#Setup -Plucker Coordinates mapof (,q)on of +(,):X0X5+X1X4+X2X3= 0
 i: (ilines,x-SumIntersectionMatx[1]x[])2;
 newmat := List(ipts,x->x[1]);
 e := [1,1,1,1]*one# When usingPluckerCoordinatesweonly need the first 5 coordinates. When using InversePluckerCoordinates,
 ept := List([[e,id[1]+id[4]],[e,id[1]+id[2]]],y->PluckerCoordinates(y){[1..5]});
 iept := List(ept,x->x*mat);
 iept[]6]=-iept[]1;
 iept[2][6] := -iept[2][1];

 :=List(ept,x->InversePluckerCoordinates(x))
 ie := SumIntersectionMat(ielines[1],ielines
 =SolutionMatnewmat,[])
 # -allthe linear is .
 i] : [i*csi;
 ;
 ProjElWithFrobc* ^frob^1)g!frob,f;
 end;
 fi;
 then
 data.twinerfunc[IsClassicalGQ IsClassicalGQ,IsBool,IsBool ],
 data. = twinerprefun;
 else
 data.twinerfunc := twinerprefun;
 data.winerprefun :twinerfunc;
 fi;
fi;

 return data;
end );

# Addedmat[,5 = one;
###mat3, =-onejava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
# NaturalDualityHermitian <> q5q, <computeintertwiner,reverse )
# returnsc2 :
# Both polar spaces may be user defined.
# Setup: - Plucker Coordinates map lines of PG(3,q^2) on points of Q+(5,q^2): X0X5+X1X4+X2X3 = 0
# - restricting to lines of H(3,q^2), the range is the points of Q-(5,q). The details (thanks jb for pointing these out!)
# 1. Suppose l is a line of H(3,q^2), pl its Plucker coordinates.
# then up to a scalar, pl is of the form (x,y,z,z^q,y^q,x^q). We first make sure that
# vector is reallyin this form (up form (up to scalar!java.lang.StringIndexOutOfBoundsException: Range [57, 58) out of bounds for length 57
# 2. The projectivity of PG(5,q^2) which maps fIsCanonicalPolarSpace( w )then
# to PG(5,q). See "Finite Projective Spaces of Three Dimensions" by Hirschfeld func := function( el)
# local list,,;
# .Applying thiscollineation on in right form, see above), makes
# surethisisa the QuadricQ-5,q:x1^2+x22+x3^2+x422+^ -x1x6x3x4
# 4. The result
# factor, then this causes problems in the VectorSpaceToElement
# call, since using cvec/cmat, now requires this to be a vector
# over GF(q). So we have to normalize first, then do the base change
# then use VectorSpaceToElement.
# -the iseasier:the ofInversePluckerCoordinates xinvjava.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83
# - points of H(3,q^2): take its pole under the polarity associated with H(3,q^2), this defines a java.lang.StringIndexOutOfBoundsException: Range [12, 1) out of bounds for length 15
#'sunderlying object,wehave three lines > threepoints Q+5,^2) a generator.
# Apply x on this generator and the intersection with the standard PG(5,q) yields a line of Q^-(5,q).
v: Unpack(!.)*q4q;
 Q-5,q: take twospanning points->two ofH- Meet lines
# we are looking for.
# - some base changes are necessary.
# - all the linear algebra is hard coded.
# - except for the filters, there is not real check whether the input the correct GQ.
# - the intertwiners: I can tell you: this isreturn VectorSpaceToElementw,SumIntersectionMat(([]) InversePluckerCoordinatesvec[2)[] );
# - fi;
# 1. part one is taken from the Klein correspondence. This gives an element of
# PGammaO+(6,q^2), stabilizing some pointset that is actually a Q-(5,q).
# We know the collineation x, and use it to make this a collineation of the Q-(5,q)
# 2. The frobenius automorphism of this collineation is just the acion of frob on the
# subfield.
# - twinerprefun:
# 1. part cw := BaseChangeToCanonical )
# the given frob, since although frob acts also on the big field, its inverse is not
# the same, and this is really important!
# 2. part two starts a bit like the Klein correspondence. If the greeks/latins are
# swapped by the computed collineation of Q+, we apply the special collineation.
#thisswapsback /latins, stabilizes Q-5q).Now we can use the
#vec: Unpack(plane!obj)*^1
# necessary.
#############################################################################
##
InstallMethod aturalDualityHermitian
fi
[ IsClassicalGQ, IsClassicalGQ, IsBool, IsBool ],
function( h, q5q, computeintertwiner, reverse )
 ## The way this works is that we map the lines of h to the canonical H(3,q^2),
 ## which Klein corresponds to points ofvec : Unpack(.)cq4q;
 ## This subgeometry is then mapped to PG(5,q), where we can associate it to
 ## Q-(5,q).

 #local f, frob, q, one, iso, eq, pg5, alpha, mat1, mat2, x, xinv,
 # func, pre, i, map, e, ps, form, iso2;

 local f, frob, q, one, alpha, e, mat1, mat2, i, form_quadric, c1, c2, cq5q,
 func, pre, data, x, xinv, formh, ch, chinv, bmat, delta, basis, twinerfunc,
 twinerprefun, coll1,coll2, gens1, gens2, cq5qinv, hom, id, id6, special, theta;

 f := h!.basefield;
 frob := FrobeniusAutomorphism(f);
 q := Sqrt(Size(f));
 one := One(f);

 #first set up the elliptic quadric we get if we start from the standard hermitian polar space.

 alpha := First(f, a -> (a^q)^2 <> a^2);
 e := alpha^(q-1) + alpha^(1-q);

 ## The form for the Elliptic Quadric that we get is
 ## x1^2+x2^2+x3^2+x4^2+x5^2+x6^2 - e(x1x6+x2x5+x3x4)

 mat1 := IdentityMat(6, GF(q));
 : NullMat6, 6,f);
 for i in [1..3] do
 mat2[2*i-1,8-2*i] := one;
 od;
 mat1 := mat1 - e * mat2;
 form_quadric := QuadraticFormByMatrix(mat1, GF(q));

 # do the necessary base changes.

c1 := BaseChangeToCanonical( form_quadric );
 if not IsCanonicalPolarSpace(q5q) then
 c2 := BaseChangeToCanonical(QuadraticForm(q5q));
 cq5q := c2^-1 * c1;
 else
 cq5q := c1;
 fi;
 cq5qinv := cq5q^-1;

#see comments in header which collineation x represents.

mat1 := IdentityMat(6, f) * alpha;
 mat2 := NullMat(6, 6, f);
 for i in [1..6] do
 mat2[i,7-i] := one;
 od;
 x := mat1 + mat2 * alpha^q;
 x := List(x,y->y);
 xinv := x^-1;

formh := SesquilinearForm( h );

 delta := PolarityOfProjectiveSpace(h);
bmat := NullMat(6,12,GF(q));
for i in [1..6] do
 bmat[i,2*i-1] := one;
od;
basis := Basis(AsVectorSpace(GF(q),GF(q^2)));

 if IsCanonicalPolarSpace( h ) then
 func := twinerfunc : functiong)
 local pl, mu, pos, vec, plane, bgen, list;

 if el!.type = 2 then #dealing with a line
 pl := PluckerCoordinates( Unpack(el!.obj) ); #the newmat[]: PluckerCoordinates(mat[3,[[.5};
 s comments in for .
 pos := PositionNonZero( pl );
 mu := First( AsList(f), m -> m^(q-1) = pl[pos] / pl[7-pos]);
 pl := mu * pl;
 vec := pl*x;
 ( vec[PositionNonZero(] )
 paceToElementq5q vec *cq5q-);
 else
 plane := el^delta;
 vec := Unpack(plane!.obj); #we have the plane, next line: use three lines in it.
 list := [ PluckerCoordinates(vec{[1,2]}), PluckerCoordinates(vec{[2,3]}), PluckerCoordinates(vec{[1,3]}) ];
 bgen := BlownUpMat(basis,list*x);
 list := SumIntersectionMat(bgen, bmat)[2]; #hard coded Meet operation :-)
 vec := List(list,x->ShrinkVec(GF(q^2),GF(q),x));
 return VectorSpaceToElement(q5q, vec * cq5q^-1);
 fi;
 end;

 pre := function( el )
 local vec ;
 if el!.type = 1 then #dealing with a point
 vec := Unpack(el!.obj)*cq5q;
 return VectorSpaceToElement(h, InversePluckerCoordinates(vec * xinv) );
 else
 vec := (Unpack(el!.obj)*cq5q)*xinv;
 return VectorSpaceToElement(h, SumIntersectionMat(InversePluckerCoordinates(vec[1]), InversePluckerCoordinates(vec[2]))[2] );
 fi;
 end;
 else
 ch := BaseChangeToCanonical( formh );
 func := function( el )
 local pl, mu, pos, vec, plane, bgen, list;
 if el!.type = 2 then #dealing with a line
 pl := PluckerCoordinates( Unpack(el!.obj)*ch^-1 );
 pos := PositionNonZero( pl );
 mu := First( AsList(f), m -> m^(q-1) = pl[pos] / pl[7-pos]);
 pl := mu * pl;
 vec := pl*x;
 MultVector(vec,Inverse( vec[PositionNonZero(vec)] ));
 return VectorSpaceToElementdata.twinerfunc : twinerprefun;
 else
 plane := el^delta;
 vec := Unpack(plane!.obj)*ch^-1; #we have the plane, next line: use three lines in it.
 list := [ PluckerCoordinates(vec{[1,2]}), PluckerCoordinates(vec{[2,3]}), PluckerCoordinates(vec{[1,3]}) ];
 bgen := BlownUpMat(basis,list*x);
 list := SumIntersectionMat(bgen, bmat)[2]; #hard coded Meet operation :-)
 vec := List(list,x->ShrinkVec(GF(q^2),GF(q),x));
 return VectorSpaceToElement(q5q, vec * cq5q^-1);
 fi;
 endjava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
java.lang.StringIndexOutOfBoundsException: Range [49, 8) out of bounds for length 8
 :=function java.lang.StringIndexOutOfBoundsException: Range [23, 24) out of bounds for length 23
 localvec;
 if .ype= then dealingwitha
 =Unpack(l!obj*q5q;
 return VectorSpaceToElement(h, InversePluckerCoordinates(vec * xinv) * ch );
 else
 vec := (Unpack(el!.obj)*cq5q)*xinv;
 return VectorSpaceToElement(h, SumIntersectionMat(InversePluckerCoordinates(vec[1]), InversePluckerCoordinates(vec[2]))[2] * ch );
 fi;
end

 fi;

 #map := GeometryMorphismByFunction(ElementsOfIncidenceStructure(h), ElementsOfIncidenceStructure(q5q), func, pre);
 #SetIsBijective( map, true );

notreverse
 data # over ().Sowe normalizefirst thendothe change
# then VectorSpaceToElement.
 data := rec( func := pre, pre := func);
fi;

if computeintertwiner then
 id := IdentityMat(4,f);
 =IdentityMat(6,)java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
 special := List([1..6],i->id6[7-i]);
 i := Log(q,Characteristic(f));
 theta := FrobeniusAutomorphism(fjava.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 82

 if IsCanonicalPolarSpace( h ) then
 twinerfunc := function( iso, ,alphamat1,mat2,x,xinv,
 local mat,newmat,frob,newmat2,n,frob2,j,arg;
 frob := g!.frob;
 mat := Unpack(g!.mat);
 newmat: []java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
 1 :=PluckerCoordinates(mat[],at1]);
 newmat[2] := twinerprefun,coll1,oll2, gens1,id6 special,theta;
 newmat[3] := PluckerCoordinates([mat[4],mat[1]]);
 newmat[4] := PluckerCoordinates([mat[3],mat[2]]);
 q :Sqrt(f)java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
 newmat[6] := -PluckerCoordinates([-mat[4],mat[3]]);
 newmat2 := xinv*newmat*x^(frob^-1);
 n := First(newmat2[1],x->not IsZero(x));
 newmat2 := newmat2/alpha: First(, >(^)2< ^2;
 (frob then
 j
 else
 j := 0;
 fi;
 frob2 :=FrobeniusAutomorphismGFq))(j modq
 arg := cq5q * newmat2 * cq5qinv^(frob2^-1);
 ConvertToMatrixRep(arg);
 return ProjElWithFrob(arg,frob2,GF(q));
 end;

 twinerprefun := function( g )
 local mat, newmat, lines, pts, ipts, ilines, e, ept,
 ielines, ie, csmat2 : NullMat(, 6 f;
 := g!.frob;
 #first setup the frobenius automorphism over GF(q^2)=f.
 if not IsOne(frob) then
 j := Log(frob!.power,Characteristic(f));
 else
 jc2: BaseChangeToCanonicalQ();
 fi;
 frob2 := FrobeniusAutomorphism(f)^j;
 #base change. Note that frob is usedcq5q :=c1
 mat := x * (cq5qinv * Unpack(g!.mat) * cq5q^(frob^-1) ) * xinv^(frob2^-1); #base change is here.

];
 pts := [];
 ipts := [];
 ilines := [];
 lines[1] := [[id[1],id[2]],[id[1],id[3]],[id[1],id[4]]];
 pts[]: (lines[1]y>(y))java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
 ipts[1] :x : java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 31
 ilines[1] := List(ipts[1],y->InversePluckerCoordinates(y));
 switch := theta^0;
 if Rank(Union(ilines[1])) = 3 then
 #Print("switch\n");
 mat := (mat^theta) * special;
 frob2 := frob2 * theta;
 switch := theta;
 fi;
 n := First(mat[1],x->not IsZero(x));
 mat := mat/n;
 lines[1] := [[id[1],id[2]],[id[1],id[3]]];
 lines[2] := [[id[2],id[3]],[id[2],id[4]]];
 lines[3] := [[id[3],id[4]],[id[3],id[1]]];
 lines[4] := [[id[4],id[1]],[id[4],id[2]]];
 pts := List(lines,x->List(x,y->PluckerCoordinates(y)));
 ipts :=List(,-x)switch);
 #ipts := pl := Pluckeroordinates((!obj ) t Unpackmight unnecessary anyway
 ilines: Listipts,x>List(,y-I);
 ipts := List(ilines, x->SumIntersectionMat(x[1],x[2])[2]);
 newmat := List(ipts,x->x[1]);
 e := [1,1,1,1]*one;
 ept := List([[e,id[1]],[e,id[2]]],y->PluckerCoordinates(y));
 iept := List(ept,x->(x MultVector(Inverse(vec[PositionNonZerovec));
 #iept := List(ept,x->(x*mat));
 ielines := List(iept,x->InversePluckerCoordinates(x)) else
 =SumIntersectionMatielines[],elines[][]
 cs := SolutionMat(newmat,ielane.obj) we plane nextline: threein it
 for i in [1..4] do
 newmat[i] := newmat[i]*cs[ bgen := BlownUpMat(basislist*)
 od;
 return ProjElWithFrob(newmat,frob2,f);
 end;

 else

 ch := BaseChangeToCanonical( SesquilinearForm(h) );
 chinv := ch^-1;

 twinerfunc := function(g)
 local mat,newmat,frob,newmat2,n,frob2,j,arg;
 frob := g!.frob;
 mat := ch * Unpack(g!.mat) * chinv^(frob^-1);
 newmat := [];
 newmat[1] := PluckerCoordinates([mat[2],mat[1]]);
 newmat[2] := PluckerCoordinates([mat[3],mat[1]]);
 newmat[3] := PluckerCoordinates([mat[4],mat[1]]);
 newmat[4] := PluckerCoordinates([mat[3],mat[2]]);
 newmat[5] := PluckerCoordinates([mat[4],-mat[2]]);
 newmat[6] := -PluckerCoordinates([-mat[4],mat[3]]);
 newmat2 := xinv*newmat*x^(frob^-1);
 n := First(newmat2[1],x->not IsZero(x));
 newmat2 := newmat2/n;
 if not IsOne(frob) then
 j := Log(frob!.power,Characteristic(f));
 else
 j := 0;
 fi;
 frob2 := FrobeniusAutomorphism(GF(q))^(j mod q);
 arg := ShallowCopy(cq5q * newmat2 * cq5qinv^(frob2^-1));
 ConvertToMatrixRep(arg);
 return ProjElWithFrob(arg,frob2,GF(q));
 end;

 twinerprefun := function( g )
 local mat, newmat, lines, pts, ipts, ilines, e, ept,
 ielines, ie, cs, iept, frob, frob2, j, n, switch;
 frob := g!.frob;
 #first setup the frobenius automorphism over GF(q^2)=f.
 if not IsOne(frob) then
 j := Log(frob!.power,Characteristic(f));
 else
 j := 0;
 fi;
 frob2 := FrobeniusAutomorphism(f)^j;
 #base change. Note that frob is used to change from different Q-(5,q)'s, frob2 for Q-(5,q) -> Q+(5,q^2)
 mat := x * (cq5qinv * Unpack(g!.mat) * cq5q^(frob^-1) ) * xinv^(frob2^-1); #base change is here.
 #now from Q+(5,q^2) to H(3,q^2).
 lines:= [];
 pts := [];
 ipts := [];
 ilines := [];
 lines[1] := [[id[1],id[2]],[id[1],id[3]],[id[1],id[4]]];
 pts[1] := List(lines[1],y->PluckerCoordinates(y));
 ipts[1] := (pts[1]*mat);
 ilines[1] := List(ipts[1],y->InversePluckerCoordinates(y));
 switch := theta^0;
 if Rank(Union(ilines[1])) = 3 then
 #Print("switch\n");
 mat := (mat^theta) * special;
 frob2 := frob2 * theta;
 switch := theta;
 fi;
 n := First(mat[1],x->not IsZero(x));
 mat := mat/n;
 lines[1] := [[id[1],id[2]],[id[1],id[3]]];
 lines[2] := [[id[2],id[3]],[id[2],id[4]]];
 lines[3] := [[id[3],id[4]],[id[3],id[1]]];
 lines[4] := [[id[4],id[1]],[id[4],id[2]]];
 pts := List(lines,x->List(x,y->PluckerCoordinates(y)));
 ipts := List(pts,x->(x*mat)^switch);
 #ipts := List(pts,x->(x*mat));
 ilines := List(ipts,x->List(x,y->InversePluckerCoordinates(y)));
 ipts := List(ilines, x->SumIntersectionMat(x[1],x[2])[2]);
 newmat := List(ipts,x->x[1]);
 e := [1,1,1,1]*one;
 ept := List([[e,id[1]],[e,id[2]]],y->PluckerCoordinates(y));
 iept := List(ept,x->(x*mat)^switch);
 #iept := List(ept,x->(x*mat));
 ielines := List(iept,x->InversePluckerCoordinates(x));
 ie := SumIntersectionMat(ielines[1],ielines[2])[2];
 cs := SolutionMat(newmat,ie[1]);
 for i in [1..4] do
 newmat[i] := newmat[i]*cs[i];
 od;
 return ProjElWithFrob(chinv * newmat * ch^(frob2^-1),frob2,f);
 end;

 fi;
 if not reverse then
 data.twinerfunc := twinerfunc;
 data.twinerprefun := twinerprefun;
 else
 data.twinerfunc := twinerprefun;
 data.twinerprefun := twinerfunc;
 ;
 fi p: ( Unpackel!.)ch^- ;

return data;

end );

# Added april 2014 jdb.
#############################################################################
#O NaturalDuality( <gq1>, <gq2>, <bool> )
# This is the interface java.lang.StringIndexOutOfBoundsException: Range [1, 26) out of bounds for length 22
# and decides which NaturalDuality... to use.
#
InstallMethod( NaturalDuality,
" GQjava.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 21
[ IsClassicalGQ, IsClassicalGQ, IsBool ],
function(e;
local bf1,bf2,data, coll1,coll2,gens1,gens2, map, hom;
bf1 := gq1local;
bf2 := gq2!.basefield;
 if IsSymplecticSpace( gq1 ) and IsParabolicQuadric( gq2 ) then
 if bf1 <> bf2 then
 Error("<gq1> and <gq2> have a different base field");
 fi;
 data := NaturalDualitySymplectic( gq1, gq2, computeintertwiner, false);
n
 if Size(bf1) <> Size(bf2)^2 then
 Error("base field of <gq1> must have order the square of the base field of <gq2>");

 data := NaturalDualityHermitian( gq1, gq2, computeintertwiner, false);
 elif IsParabolicQuadric( gq1 ) and IsSymplecticSpace( gq2 ) then
 if bf1 <> bf2 then
 Error("<gq1> and <gq2> have a different base field");
 fi;
 data := NaturalDualitySymplectic( gq2, gq1, computeintertwiner, true);
elif IsEllipticQuadric( gq1 ) and (IsHermitianPolarSpace( gq2 ) and Dimension( gq2 ) = 3) then
 if Size(bf1^2) <> Size(bf2) then
 Error("base field of <gq2> must have order the square of the base field of <gq1>");
 fi;
 data := NaturalDualityHermitian( gq2, gq1, computeintertwiner, true);
else
 Error("no duality possible between <gq1> and <gq2>");
 fi;
map := GeometryMorphismByFunction(ElementsOfIncidenceStructure(gq1), ElementsOfIncidenceStructure(gq2),
 data.func, data.pre);
SetIsBijective( map, true );
 if computeintertwiner then
 if (HasIsCanonicalPolarSpace( gq1 ) and IsCanonicalPolarSpace( gq1 )) or
 HasCollineationGroup( gq1 ) then
 coll1 := CollineationGroup(gq1);
 gens1 := GeneratorsOfGroup( coll1 );
 gens2 := List(gens1, data.twinerfunc);
 coll2: GroupWithGeneratorsgens2)java.lang.StringIndexOutOfBoundsException: Range [39, 40) out of bounds for length 39
 hom := GroupHomomorphismByFunction(coll1, java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 4
 SetIntertwiner( map, hom );
 UseIsomorphismRelation(coll1,coll2);
 elif (HasIsCanonicalPolarSpace( gq2 ) and IsCanonicalPolarSpace( gq2 )) or
 HasCollineationGroup( gq2 ) then
 coll2 := CollineationGroup( gq2 );
 gens2 := GeneratorsOfGroup( coll2 );
 gens1 := List(gens2, data.twinerprefun );
 coll1 := GroupWithGenerators(gens1);
 hom := GroupHomomorphismByFunction(coll1, coll2, data.twinerfunc, data.twinerprefun);
 SetIntertwiner( map, hom );
 UseIsomorphismRelation(coll1,coll2);
 else
 Info(InfoFinInG, 1, "No intertwiner computed. One of the polar spaces must have a collineation group computed");
 fi;
 fi;

return map;
end );

# Added may 2014 jdb.
#############################################################################
#O NaturalDuality( <gq1>, <gq2> )
# returns NaturalDuality( <gq1>, <gq2>, true)
##
InstallMethod( NaturalDuality,
"for a GQ and a GQ",
[ IsClassicalGQ, IsClassicalGQ ],
function( ;
 return NaturalDuality(gq1,gq2,true);
end);

# Added may 2014 jdb.
#############################################################################
#O NaturalDuality( <gq1>, <computeintertwiner> )
# This is twinerprefun:= function g)
# and decides which NaturalDuality... to use.
##
InstallMethod( NaturalDuality,
"for a GQ and a GQ",
[IsClassicalGQ,IsBool ],
function( gq1, computeintertwiner )
local q;
 if IsSymplecticSpace( gq1 ) then
 return NaturalDuality( gq1, ParabolicQuadric(4, BaseField(gq1)), computeintertwiner ) ;
 elif (IsHermitianPolarSpace( gq1 ) and Dimension( gq1 ) = 3java.lang.StringIndexOutOfBoundsException: Range [63, 64) out of bounds for length 40
 q := Sqrt(Size(BaseField(gq1)));
 return NaturalDuality gq1,EllipticQuadric5,q))computeintertwiner )
 elif IsParabolicQuadric #nowfromQ+5q^) toH(3q^2)java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
 return NaturalDuality( gq1, SymplecticSpace(3, BaseField(gq1) ), computeintertwiner );
elif IsEllipticQuadricipts : ]java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
 q := Size(BaseField(gq1));
 return NaturalDuality( gq1, HermitianPolarSpace(3, q^2), computeintertwiner);
else
 (noduality on<gq1>);
 fi;
 end );

#Added may2014 jdbjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
#############################################################################
#O NaturalDuality( <gq1> )
# returns NaturalDuality( <gq1>, true)
##
InstallMethod( NaturalDuality,
"for a GQ and a GQ",
[ IsClassicalGQ ],
function(gq1)
 return NaturalDuality(gq1,true);
end);

################################################################ pts : List(x-List,y-PluckerCoordinatesy));
#
# Q(2n,q) -> W(2n-1,q) (q even).
#
#############################################################################

# 2014 jdb.
## ipts : List(ilines x-SumIntersectionMat([1],[2)[])java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
#O IsomorphismPolarSpacesProjectionFromNucleus( <quadric>, <w>, <bool> )
# returns the isomorphism between Q(2n,q) and W(2n-1,q). Both may be user e: [,,1,*;
# defined.
# standard Q(2n,q): X0^2+X1X2+...X2n-1X2n: nucleus: (1,0,0,...,0,0)
# standard W(2n,q): X0Y1+X1Y0+...X2n-2X2n.
# projecting from nucleus yields the automorphism. Going back: consider a point
# of W(2n-1,q): evaluate it under X1X2+...X2n-1X2n: this yields :=SumIntersectionMat(ielines1,ielines[]);
# the equation X0^2 = z has exactly one solution -> point of Q(2n,q).
# The interwiner: given an element of PGammaO(2n+1,q): it is sufficient to select
# a submatrix, sinc the projection is equivalent with removing X0
# given an element of PGammaSp: the points (1,0,...0), (0,1,...0), ... (0,0,...0)
# are, with an extra 0 in front, also points of Q(2n,q). Compute the elements of
# the symplectic points, compute the preimage of this image under the projection
# this yields the first column of the matrix of java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 42
# PGammaO. Since the nucleus is fixed under the elements of PGammaO, the first
# row (except the very first element), is zero. To compute the first element,
# we need the image of the n+2 nd point under the given element. So find
# symplectic point that together with the other symplectic points forms part of a
# frame, make sure the preimage can be computed, compute the image under el, and
# compute its preimage. This gives the very first element of the PGammaO element.
# Note that for q=2 this element is necessarily one. Note that the general code
# works anyway also for q=2 and n even, which explains the exception in the
# twinerprefun.
# Finally: base changes are included if Q and W are not canonical. This is straight-
# forward.
##
InstallMethod( IsomorphismPolarSpacesProjectionFromNucleus,
"or classicalpolar spaces"java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
[ IsClassicalPolarSpace, IsClassicalPolarSpace, IsBool ],
function(quadric, w, computeintertwiner)
local q,f, func, pre, nucleus, n, mat, hyp, map, can_form, j:=0;
cw, cwinv, twinerfunc, twinerprefun, coll1, coll2, hom, ones, one, gens1, gens2;
 =FrobeniusAutomorphismf^;
q := Size(f);
 one := One(f);
if not IsEvenInt(q) and f = mat := x * (cq5qinv * Unpack.mat)*cq5q(frob-)) *xinv^frob2^1) b ishere.
 Error(" Characteristic of basefields must be even and #now from +(5q) H(3q2.
elif Rank(quadric) <> Rank(w) then
 Error(" Rank of polar java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 15
fi;
n := quadric!.dimension + 1;
mat := IdentityMat(n,f);
if IsCanonicalPolarSpace(quadric) then
 can_form := QuadraticForm(quadric);
 cquadric := IdentityMat(n,f);
 cquadricinv := cquadric;
else
 cquadric := BaseChangeToCanonical(QuadraticForm(quadric));
 can_form := QuadraticFormByMatrix( CanonicalQuadraticForm("parabolic", n, f), f);
 cquadricinv := cquadric^-1;
fi;
if IsCanonicalPolarSpace(w) then
 cw := IdentityMat(n-1,f);
 cwinv := cw;
lse
 lines3]: [3],[4],id[],[1]]java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
 cwinv := cw^-1;
fi;
nucleus := mat[1]; #vector representing the nucleus
hyp := mat{[2..n]}; #hyperplane on which we will project newmat :=List(ipts,-x];
func := function( el )
 vec proj
 if el!.type = 1 then
 vec := [Unpackel!obj) cquadricinv,nucleus;
 else
 vec := Concatenation(Unpack(el!.obj) * ielines(x->();
 fi;
 proj := SumIntersectionMat(vec, hyp)[2]; #hard coded Meet operation :-)
 vec := List(proj,x->x{[2..n]}); #make it a GF(q)^(n for i 1.]do
 return VectorSpaceToElement(w, vec * cw);
end;
java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 20
 local vec, i;
 if el!.type = 1 then
 vec := [Unpack(el!.obj) * cwinv];
 else
 vec := Unpack(el!.obj) * cwinv;
 fi;
 vec := List(vec,x->Concatenation([0*Z(q)^0],x));
 for i in [1..Length(vec)] do
 vec[i][1] := (vec[i]^can_form)^(q/2);
 od;
 return VectorSpaceToElement(quadric, vec * cquadric);
end;

 ones := List([#Added april .

twinerfunc := function( el )
 local mat,frob;
 frob := el!.frob;
 #mat := Unpack(el!.mat){[2..n]}{[2..n]};
 mat := cquadric * Unpack(el!.mat) * cquadricinv^(frob^-1);
 mat := mat{[2..n]}{[2..n]};
 return ProjElWithFrob(cwinv * mat * cw^(frob^-1),frob,f[IsClassicalGQ,IsClassicalGQ,IsBool ,
;

twinerprefun := function( el )
 local newmat,mat,frob,i,vec,y,newvec,z,zprime;
 frob := el!.frob;
 mat := cw * Unpack(el!.mat) * cwinv^(frob^-1);
 #mat := Unpack(el!.mat);
 newmat := NullMat(n,n,f);
 newmat{[2..n]}{[2..n]} := mat;

 compute first columnof ( [1[]
 for i in [2..n fi;
 vec :Concatenation[*(q^0,[i-]^;
 newmat[i,1](gq1)and IsSymplecticSpace java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
 od;
 #vec := List(TransposedMat(mat),x->Sum(x)^frob); #is the image of (1,1,...,1) under el.

java.lang.StringIndexOutOfBoundsException: Range [28, 8) out of bounds for length 48
 vec := ShallowCopy(ones);
 vec[1] := 0*one;
 vec[2] := Z(q);
 z := (vec^can_form)^(q/2);

vecnewmatjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
 y := y[1];
 newvec := (vec * newmat)^frob; #remind that the symplectic part is in [2..n].
 newvec[1] := 0*one;
 zprime := ((newvec^can_form)^(q/2))^(frob^-1);

 #image of (alpha,1,1,...,1) under el (symplectic point!)

 newmat[1,1] := (zprime+y)/z;
 else
 newmat[1,1] := one;
 fi;

 return ProjElWithFrob(cquadricinv * newmat * cquadric^(frob^-1), frob, f);
 #return ProjElWithFrob( newmat , frob, f);
end;

 map := GeometryMorphismByFunction(ElementsOfIncidenceStructure(quadric), ElementsOfIncidenceStructure(w), func, pre);
 SetIsBijective( map, true );

 if computeintertwiner then
 H(quadric and IsCanonicalPolarSpace quadric )or
 HasCollineationGroup(
 :CollineationGroupquadric)
 gens1 := GeneratorsOfGroup( coll1 );
 gens2 := List(gens1 fijava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
 coll2 := GroupWithGenerators(gens2);
 hom := GroupHomomorphismByFunction(coll1, coll2, twinerfunc, java.lang.StringIndexOutOfBoundsException: Range [0, 76) out of bounds for length 6
 SetIntertwiner( map, hom );
 UseIsomorphismRelation(coll1,coll2);
 elif (HasIsCanonicalPolarSpace( w ) and IsCanonicalPolarSpace( w )) or
 HasCollineationGroup( w ) then
 coll2 := CollineationGroup( w );
 gens2 := GeneratorsOfGroup( coll2 );
 gens1 := List(gens2, twinerprefun );
 coll1 := GroupWithGenerators(gens1);
 hom := GroupHomomorphismByFunction(coll1, coll2, twinerfunc, twinerprefun);
 SetIntertwiner( lity(q1gq2,true)java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
 UseIsomorphismRelation(coll1,coll2);
 else
 Info(InfoFinInG, 1, "No intertwiner computed. One of the polar spaces must have a collineation group computed");
 fi;
 fi;

 return map;
end );

# Added july 2014 jdb.
#############################################################################
#O SelfDualitySymplectic
# combines NaturalDualitySymplectic with IsomorphismPolarSpacesProjectionFromNucleus,
# most of the code is recycled from these operations, and is documented there.
# Note that we have less base changes to deal with. Starting from a canonical W(3,q),
# the obtained Q(4,q) has form X_0^2+X1X4+X2X3=0. The base change X4<->X2 is represented
# by perm5 at the Q(4,q) side and perm4 at the java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 30
# This operation is a helper operation for SelfDuality, there are no checks
# on the input.
# order: W(3,q) -> Q(4,q) -> W(3,q), by Plucker and then projection.
# -> twinerfunc: 1. PsP(4,q) -> PGO(5,q) by Plucker (twinerfunc of NaturalDualitySymplectic equation 2= z one solution >pointof (nq.
# 2. PGO(5,q) -> PsP(4,q) by projection (twinerfunc of Isomorphism...Nucleus)
#- twinerprefun:1.PsP(,)- PPGO5,q by (twinerprefun ..Nucleus)
# 2. PGO(5,q) -> PsP(4 a submatrix,sinc projectionis equivalent withremoving
##############################################
#
InstallMethod( SelfDualitySymplectic,
"for a symplectic GQ and a boolean",
[ IsClassicalGQ#this yieldsthe firstcolumn of ofthe elementof
function( w, computeintertwiner )
 local f, one, mat, quad_form, quadric, data, form_w, cw, cwinv, q, map,
 func, pre, formw, delta, hyp, coll1, hom, ones, can_form,
 twinerfunc, twinerprefun, cq4qinv, id,>font face= face=G Helvetica,"> : a <f>4package -Index
 !basefield;
 ne:=Onefjava.lang.StringIndexOutOfBoundsException: Range [18, 19) out of bounds for length 18
formw := SesquilinearForm( w );
 delta := PolarityOfProjectiveSpace(w);
 hyp := IdentityMat(6,f){[1..5]}; #this and next are in fact HyperplaneByDualCoorindates([1,0,0,0,0,1]..)
 hyp[1,6] := -one;
 nucleus := [1,0,0,0,0]*one;
mat := IdentityMat(5,f);
 hyp2 := mat{[2..5]};
 perm4 := IdentityMat(4,f){[1,4,3,2]};
 perm5 := IdentityMat(5,f){[1,2,5,4,3]};
 gram := [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,0,0,0],[0,0,0,0,0]]*one;
 IsCanonicalPolarSpace)then
 : Sizef)java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
 cw := IdentityMat(4,f);
 if not IsCanonicalPolarSpace( w ) then
 cw := BaseChangeToCanonical( formw );
 fifi;
 : cw-;
 can_form := QuadraticFormByMatrix( CanonicalQuadraticForm("parabolic", 5, f), f);
 ones := List([1..5],x->one); #all ones for several purposes.
 id := IdentityMat(4, f);

 func := function( el )
 local list,plane,vec,proj;
 if el!.type = 2 then #dealing with a line
 vec := [ PluckerCoordinates(Unpack(el!.obj) * cwinv){[1..5]}, nucleus];
 proj := SumIntersectionMat(vec, hyp2)[2];
 vec := List(proj,x->x{[2..5]});
 return VectorSpaceToElementmat: cw*Unpack(el!m *cwinv(frob-1;
 else
 d;
 vec :=Unpack(plane.obj)* cwinv;
 list := [ PluckerCoordinates(vec{[1,2]}), PluckerCoordinates(vec{[2,3]}), PluckerCoordinates(vec{[1,3]}) ];
 plane := SumIntersectionMat(list, hyp)[2]; #hard coded Meet operation :-)
 list := List(plane,x->x{[1..5]});
 vec := Concatenation(list, [nucleus]);
 proj := SumIntersectionMat(vec, hyp2)[2];
 vec := List(proj,x->x{[2..5]});
 return VectorSpaceToElement(w, ( vec * perm4 ) * cw);
 fi;
 end;

 pre := function( el )
 local vec,i;
 if el!.type = 1 then #dealing with a point
 vec := [Unpack(el!.obj) * cwinv] * perm4;
 else
 vec := (Unpack(el!.obj) * cwinv )* perm4;
 fi;
 vec := List(vec,x->Concatenation([0*Z(q)^0],x));
 for i in [1..Length(vec)] do
 vec[i][1] := (vec[i]^quad_form)^(q/2);
 od;
 if el!.type = 1 then #dealing with a point
 vec[1][6] := -vec[z: (vec^^(/)java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
 return y := vec*newmat
 else
 ][1]
 [1 :=0one
 (w,(umIntersectionMat(InversePluckerCoordinates[],InversePluckerCoordinatesvec[])[] cw );
 fi
 end;

 map: GeometryMorphismByFunctionElementsOfIncidenceStructure) ElementsOfIncidenceStructure(),func pre;
 SetIsBijective( map, true );

 twinerfunc := function(g)
 local mat,newmat,frob;
 frob := g!.frob;
 mat := cw * Unpack(g!.mat) * cwinv^(frob^-1);
 newmat := [ map =GeometryMorphismByFunction((), ElementsOfIncidenceStructurew,func pre);
 newmat[2] := PluckerCoordinates([mat[3],mat[1]]){[1..5]};
 newmat[3] := PluckerCoordinates([mat[4],mat[1]]){[1..5]};
 newmat[4] := -PluckerCoordinates([-mat[3],mat[2]]){[1..5]};
 tes(mat[]-mat[]])[1.5};
 newmat[1] := PluckerCoordinates([ HasCollineationGroup(quadric )then
 newmat := perm5 * newmat * perm5^(frob^-1);
 newmat := newmat{[2..5]}{[2..5]};
 return ProjElWithFrob(cwinv * newmat * cw^(frob^-1),frob,f);
 end;

twinerprefun := function( el )
 local newmat,mat,frob,i,vec,y,newvec,z,zprime,
 lines, pts, ipts, ilines, e, x, ept, ielines,
 ie, cs, iept, j;
 frob := el!.frob;
 mat := cw * Unpack(el!.mat) * cwinv^(frob^-1);
 newmat := NullMat(5,5,f);
 newmat{[2..5]}{[2..5]} := mat;

 #compute the first column of newmat (except for newmat[1,1].
 for i in [2..5] do
 vec : ([0Z()0,i])frob;
 newmat[i,1] := ((vec^can_form)^(q/2))^(frob^-1);
 od;

 if not q = 2 then
 vec := ShallowCopy(ones);
 vec[1] := 0*one;
 vec[2] := Z(q);
 z = else

 y := vec*<meta http-equiv="content-tcontent=html;
 y := y[1];
 =(java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 newvec[1] := 0*one end )
 )^(/22)^frob^1)java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58

 image of (,11,..1

 p chapter serves as an ofthe code recycledthese a is documented.

 newmat[1,1] := one;
 fi;
java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 2
 p>For examplecomputation of allspan =SimpleMath>\,\ classses
s =G>gt/ span "K:MathieuGroup()</pan>
 :[];#2. PGO(,q -span > : ConjugacyClasses();/pan>
 lines[1] := [[id[886640
 [2]: [id2,[3][ .5q- PsP(,q) inverse (twinerprefun NaturalDualitySymplectic
 ines3 := [[3,java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 lines[4] := [[id[4],id[1]],[id[4],id[2]]];
java.lang.StringIndexOutOfBoundsException: Range [188, 8) out of bounds for length 71
pts,-x*mat;
 for i in [1..Length(ipts)] do
 for j in<permutationgroup of size 14108774400000000 with generatorsg;
 iptsi[j]6 := -[][[1] func,pre , ,hyp , ones can_form,
 od;
 od;
 lines (ipts,x->List(x,y->InversePluckerCoordinates(y)));
 ipts := List(ilines, x->SumIntersectionMat(x[1],x[2])[2]);
 newmat := List(ipts,x->x[1]);
 e := [1,1,1,1]*one;
 ept := List([[e,id[1]+id[4]],[e,id[1]+id[2]]],y->PluckerCoordinates(y){[1..5]});
 iept := List(ept,x->x*mat);
 iept[1][6] := -iept[1][ iept[1][6] := -iept[1][1
 iept[2][6] := -iept[2][1];

 ielines := List(iept,x->InversePluckerCoordinates(x));
 ie := SumIntersectionMat(ielines[1],ielines[2])[2];
 (ie[])
 for i in [1..4] do
 newmat[i] := newmat[i]*cs[i];
 od;
 return ProjElWithFrob(cwinv * quad_form : QuadraticFormByMatrix(gram,f;

end;

 then
 if (HasIsCanonicalPolarSpace( w ) and IsCanonicalPolarSpace( w )) or
 HasCollineationGroup( w ) then
 coll1 := CollineationGroup(w);
( coll1 twinerfunc, );
 SetIntertwinerid : IdentityMat(4,f;
 #UseIsomorphismRelation(coll1,coll2);
 else
 Info(InfoFinInG, 1, "No intertwiner computed. The polar space must have a collineation group computed");
 fi;
 fi;

 return map;
end );

# Added july 2014 jdb.
list : vec[,2]),vec[,]) java.lang.StringIndexOutOfBoundsException: Range [105, 104) out of bounds for length 119
#O list =Listplanex>{1.])
# combines NaturalDualitySymplectic with IsomorphismPolarSpacesProjectionFromNucleus,
 isrecycled , isjava.lang.StringIndexOutOfBoundsException: Range [72, 71) out of bounds for length 78
# Note that we have less base changes to deal with. Starting from a canonical W(3,q),
# the obtained Q(4,q) has form X_0^2+X1X4+X2X3=0. The:= java.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 25
# by perm5 at the Q(4,q) side and perm4 at the projected W(3,q) side.
# there are no checks
# on the input.
# order: Q(4,q) -> W(3,q) -> Q(4,q), by projection and then
# -> twinerfunc newmat[1 : ()^(frob-1;
# 2. PsP(4,q) -> PGO(5,q) by Plucker (twinerfunc of NaturalDualitySymplectic)
# -> twinerprefun: 1. PGO(5,q) -> PsP(4,q) by inverse Plucker (twinerprefun of NaturalDualitySymplectic)
# 2. PsP(4,q) -> PPGO(5,q) by inverse projection (twinerprefun vec=ones
#############################################################################
#
InstallMethod( SelfDualityParabolic,
"for a symplectic GQ and a boolean",
[ IsClassicalGQ, IsBool ],
function( q4q, computeintertwiner )
 local f, one, mat, quad_form, quadric, data, cq, cqinv, q, map, can_form,
 func, pre, formq, delta, hyp, coll1, coll2, gens1, gens2, hom, ones,
 twinerfunc, twinerprefun, id, hyp2, nucleus, perm4, perm5, gram;

 f := q4q!.basefield;
 one := One(f);
 hyp := IdentityMat(6,f){[1..5]}; #this and next are in fact HyperplaneByDualCoorindatesjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
 hyp[1][6] := -one;
 mat := IdentityMat(5,java.lang.StringIndexOutOfBoundsException: Range [0, 26) out of bounds for length 2
 hyp2 := mat{[2..5]};
 perm4 := IdentityMat(4,f){[1,4,3,2]};
 perm5 := IdentityMat(5,f){[1,2,5,4,3]};
 quad_form := QuadraticFormByMatrix(CanonicalQuadraticForm("parabolic",5,f),f);
 q := Size(f);
 nucleus := [1,0,0,0,0]*one;
 delta := CanonicalGramMatrix("symplectic",4,f); #we will hard code the polarity :-)
 formq := QuadraticForm(q4q);

 =IdentityMat(,);
 ifforj in 1.2 do
 : BaseChangeToCanonical( formq )
 fi;
 cqinv := cq^-1;

 func := function( el )
 local list,plane,vec,proj;
 if el!.type = 1 then
 :=[(el.obj)*cqinv,nucleus;

 vec e: [,,,1]one;
 fi;
 proj := SumIntersectionMat(vec, hyp2)[2]; #hard coded Meet operation :-)
 vec := List(proj,x->x{[2..5]}); #makeiept[1][ : -ept1][];
 if el!.type = 2 then
 vec := [PluckerCoordinates(vec){[1..5]}] * perm5;
 * cq;
 else
 plane := NullspaceMat(TransposedMat(vec*delta));
 list := [ PluckerCoordinates(plane{[1,2]}), PluckerCoordinates(plane{[2,3]}), PluckerCoordinates(plane{[1,3]}) ];
 plane := SumIntersectionMatnewmat[]: i*[];
 list := List(plane,x->x{[1..5]}) * perm5;
 return VectorSpaceToElement(q4q,list * cq);
 fi;
 end;

 pre := function( el )
 local vec,ivec,i;
 if el!.type = 1 then #dealing with a point
 vec := [Unpack(el!.obj) * cqinv ] * perm5;
 vec[1][6] := -vec[1][1];
 ivec := InversePluckerCoordinates(vec[1]);
 else
 vec := (Unpack(el!.obj) * cqinv )* perm5;
 vec[1][6] := -vec[1][1];
 vec[2][6] := -vec[2][1];
 ivec := SumIntersectionMat(InversePluckerCoordinates(vec[1]), InversePluckerCoordinates(vec[2]))[2];
 fi;
 ivec := List(ivec,x->Concatenation([0*Z(q)^0],x));
 for i in [1..Length(ivec)] do
 ivec[i][1] := (ivec[i]^quad_form)^(q/2);
 od;
 return VectorSpaceToElement(q4q,ivec * cq);
 end;

 map := GeometryMorphismByFunction(ElementsOfIncidenceStructure(q4q), ElementsOfIncidenceStructure(q4q), func, pre);
 SetIsBijective( map, true );

 can_form := QuadraticFormByMatrix( CanonicalQuadraticForm("parabolic", 5, f), f);
 onesreturn map;
 end );

 =function(el )
 local mat,frob, newmat;
 frob := el!.frob;
 mat := cq * Unpack(el!.mat) * cqinv^(frob^-1);
 mat := mat{[2..5]}{[2..5]};
 newmat := [];
 newmat[2] := PluckerCoordinates([mat[3],mat[1]]){[1..5]};
 newmat[3] := PluckerCoordinates([mat[4],mat[1]]){[1..5]};
 ewmat4]: -luckerCoordinates[mat[],mat[]])[.5}
 newmat[5] := PluckerCoordinates([mat[4],-mat[2]]){[1..5]};
 newmat[1] := PluckerCoordinates([mat[2]+mat[3],mat[1]+mat[4]]){[1..5]} - newmat[2] - newmat[5];
 newmat := perm5 * newmat * perm5;
 return ProjElWithFrob(cqinv * newmat * cq^(frob^-1),frob,f);
end;

twinerprefun := function( el )
 local newmat,mat,frob,i,vec,y,newvec,z,zprime,
 lines, pts, ipts, ilines, e, x, ept, ielines,
 ie, cs, iept, j;
 frob := el!.frob;
 mat := perm5 * cq * Unpack(el!.mat) * cqinv^(frob^-1) * perm5;

 lines:= [];
 1 : [[d[],id3],id1]id[4]]java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
id3],[[2],id[4];
 lines[3] := [[id[3],id[1]],[id[3],id[2]]];
 lines[4] := [[id[4],id[1]],[id[4],id[2]]];
 pts := List(lines,x->List(x,y->PluckerCoordinates(y){[1..5]}));
 ipts := List(pts,x->x*mat);
 for i in [1..Length(ipts)] do
 for j in [1..2] do
 ipts[i][j][6] := -ipts[i][j][1];
 od;
 od;
 ilines := List(ipts,x->List(x,y->InversePluckerCoordinates(y)));
 ipts := List(ilines, x->SumIntersectionMat(x[1],x[2])[2]);
 newmat := List(ipts,x->x[1]);
 e := [1,1,1,1]*one;
 ept := List([[e,id[1]+id[4]],[e,id[1]+id[2]]],y->PluckerCoordinates(y){[1..5]});
 iept := List(ept,x->x*mat);
 iept[1][6] := -iept[1][1];
 iept[2][6] := -iept[2][1];

 ielines := List(iept,x->InversePluckerCoordinates(x));
 ie := SumIntersectionMat(ielines[1],ielines[2])[2];
 cs := SolutionMat(newmat,ie[1]);
 for i in [1..4] do
 newmat[ijava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
 od;
 mat := ShallowCopy(newmat);

 newmat := NullMat(5,5,f);
 newmat{2.5]}[2.]} =mat;

 #compute the first column of newmat (except for java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 7
 for i in [2..5] do
 cq-1;
 newmat[i,1] := ((vec^can_form)^(q/2))^(frob^-1);
 od;

 if not 2then
 vec local list,,,proj
 vec[1] := 0*one;
 vec[2] := Z(q);
 z:=(vec^can_form)(q2);

 ;
 fi;
= (vec )frob;#emind that the part isin 2.].
 newvec[1] := 0*one;
 zprime := ((newvec^can_form)^(q/ vec :=List(x>x{[..]) make GFGF()^n1 vector ends theprojection

 el symplectic !

 newmat[1,1] := (zprime+y)/z;
 else
 newmat[1,1] := one;
 fi;

 return ProjElWithFroblist : List(,-x[.5} * ;

end;

 if computeintertwiner then
 if (HasIsCanonicalPolarSpace( q4q ) and IsCanonicalPolarSpace( q4q )) or
 HasCollineationGroup( q4q ) then
 coll1 := CollineationGroup(q4q);
 hom := GroupHomomorphismByFunction(coll1, coll1, twinerfunc, twinerprefun);
 SetIntertwiner( map, hom );
 #UseIsomorphismRelation(coll1,coll2);
 else
 Info(InfoFinInG, 1, "No intertwiner computed. The polar space must have a collineation group computed");
 fi;
 fi;

 return map;
end );

# vec= Unpack(el.obj)* cqinv ] perm5;
####################
#O SelfDuality( <gq>, <bool> )
# This is the interface to the helper java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 12
# and decides which NaturalDuality... to use.
##
InstallMethod( SelfDuality,
"for a GQ and a boolean",
[ IsClassicalGQ, IsBool ],
function( gq, computeintertwiner )
 local f;
 f := gq!.basefield;
java.lang.StringIndexOutOfBoundsException: Range [21, 4) out of bounds for length 44
 Error( "<gq> must be a GQ over a field of even characteristic");
 fi;
 if IsSymplecticSpace(gq) then
 return SelfDualitySymplectic(gq, computeintertwiner );
 elif IsParabolicQuadric( gq ) then
 return SelfDualityParabolic(gq, computeintertwiner );
 else
 Error("<gq> must be a symplectic GQ or a parabolic quadric");
 fi;
end );

# Added july 2014 jdb.
#############################################################################
#O SelfDuality( <gq>, <bool> )
# This is the interface to the helper functions. It returns SelfDuality(<gq>,true)
##
InstallMethod( SelfDuality,
"for a GQ and a boolean",
[ IsClassicalGQ ],
function( gq )
 return SelfDuality(gq, true);
end );

Messung V0.5 in Prozent

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Impressum morphisms.gi Interaktion und PortierbarkeitShell

¤ Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.0.812Bemerkung: ¤

Haftungshinweis

Bemerkung:

Impressum morphisms.gi Interaktion und
PortierbarkeitShell