Impressum morphisms.gi
Interaktion und
PortierbarkeitShell
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#############################################################################
##
## 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
eptwe.
# CHECKED 27/ 09/ 11 jdb
###### </tem>
#O GeometryMorphismByFunction( <els1>, <els2>, <fun>, <bool>, <prefun> )
##
InstallMethod( GeometryMorphismByFunction,
"for two times a collection
[ IsAnyElementsOfIncidenceStructure, IsAnyElementsOfIncidenceStructure,
,IsBool, IsFunction ,
function( els1, els2, fun, bool, prefun )
java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
morphism := MappingByFunction(els1, java.lang.StringIndexOutOfBoundsException: Rang e [17, 16) out of bounds for length 69
SetFilterObj( morphism, java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 50
return morphism;
end );
# CHECKED 27/09/11 jdb
######################################
#O GeometryMorphismByFunction( <els1>, <els2>, <fun>, <prefun> )
##
InstallMethod( GeometryMorphismByFunction,
"for two times a collection of Otherwise if <A>G</A> is class-wisepreserving
[ IsAnyElementsOfIncidenceStructure, of the permutation group induced on the stored respected partition.
IsFunction, IsFunction ],
function( els1, els2, fun, inv )
local morphism;
morphismIf<><A - orderp,returnthesize
SetFilterObj( morphism, IsGeometryMorphism );
return morphism;
end );
# permutation groupi of thestored
################################################## hich isobtained by splittingeachresidueclass
#O GeometryMorphismByFunction( <els1>, <els2>, <fun> )
#
InstallMethod( GeometryMorphismByFunction,
"for two times java.lang.StringIndexOutOfBoundsException: Range [0, 17) out of bounds for length 9
[ java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 8
IsFunction ],
function( <C>Size(<A<A>)<C>(of rcwa )
local morphism;
morphism := MappingByFunction(els1, els2, fun);
SetFilterObj( morphism, IsGeometryMorphism );
return morphism;
end );
###################################
# Display methods
##############################Ijava.lang.StringIndexOutOfBoundsException: Range [8, 9) out of bounds for length 8
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( <em>, <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>, <em> )
##
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( <em>, <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( <em>, <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( <em>, <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
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),
func, false, pre );
SetIsInjective( map, true );
else
Error("Polar spaces are not compatible");
fi;
return map;
end );
# CHECKED 28/09/11 jdb (and added a check that basefields of <ps1> and <ps2> are equal.)
#cmat changed 20/3/14.
#############################################################################
#O java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 40
# returns ,d1basvecsjava.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
# polar spaces of the same typeifIsIntt
# computed.
##
( IsomorphismPolarSpaces
"for two similar polar spaces and a boolean",
[ IsClassicalPolarSpace, IsClassicalPolarSpace, IsBool ],
function , ps2 computeintertwiner )
local form1, java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 4
coll1, coll2, gens1, gens2, x, y, java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 0
ty1=PolarSpaceType )java.lang.StringIndexOutOfBoundsException: Range [31, 32) out of bounds for length 31
ty2 := PolarSpaceType( ps2 );
= "arabolic"andty2= symplectic"
fi;
r1 := Rank(ps1);
r2 := Rank(ps2);
if ty1 <> ty2 or r1 <> r2 then
Error("Polar spaces are of different type or of different rank");
fi;
f := ps1!.basefield;
if f <> ps2!.basefield then
Error( "<ps1> and <ps2> must be polar spaces over the same field");
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 );
fi;
c1 := BaseChangeToCanonical( form1 );
c2 := BaseChangeToCanonical( form2 );
change := NewMatrix(IsCMatRep, ps1!.basefield, ps1!.dimension+1,c1^-1 * c2); #cmat change here
#ConvertToMatrixRep(change, f);
invchange := Inverse(change);
if ty1 = "hermitian" then
change^ompanionAutomorphism)java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
invchange =invchange^ompanionAutomorphism(;
fi; q,trpg2,at1,mat2;
func := function(x)
return VectorSpaceToElement(ps2, ShallowCopy(x!.obj) * change q:basis!.java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
inv := function(x Error(The basisandtheprojectivearenotcompatible!");
return VectorSpaceToElement(ps1, ShallowCopy(x!.obj) * invchange);
end;
map := GeometryMorphismByFunction(ElementsOfIncidenceStructure(ps1),
ElementsOfIncidenceStructure(ps2),
func, inv);
# map from gens2 to gens1
twinerprefun := function( x )
local y;
y := MutableCopyMat( x!.mat );
#return ProjElWithFrob( change * y * invchange^x!.frob, x!.frob, f );
return ProjElWithFrob( change * y * invchange^(x!.frob^-1), x!.frob, f );
end;
# map from gens1 to gens2
twinerfunc := function( y )
local x;
x := MutableCopyMat( y!.mat );
#return ProjElWithFrob( invchange * x * change^y!.frob, y!.frob, #return ProjElWithFrob( invchange * x * change^y!.frob, y!.frob, f
return (invchange x *^y!.frob^1, y!frob ;
end;
if computeintertwiner then
if (HasIsCanonicalPolarSpace( ps1 ) and IsCanonicalPolarSpace( ps1 )) or
HasCollineationGroup( ps1 ) then
coll1 := CollineationGroup(ps1);
gens1 := GeneratorsOfGroup( coll1 );
gens2 := List(gens1, twinerfunc);
coll2 := GroupWithGenerators(gens2);
hom := GroupHomomorphismByFunction(coll1, coll2, twinerfunc, twinerprefun);
SetIntertwiner( map, hom );
UseIsomorphismRelation(oll1coll2)
e(asIsCanonicalPolarSpaceps2 ) andIsCanonicalPolarSpace( ps2 )) or
HasCollineationGroup( ps2 ) then
coll2 =CollineationGroup ps2 );
gens2 := BlownUpProjectiveSpaceBySubfield( <subfield>, <pg> )
gens1 := List(gens2, twinerprefun );
coll1 := # blows up a projective space a space field reduction wr. basis
hom := GroupHomomorphismByFunctionI( BlownUpProjectiveSpaceBySubfield,
SetIntertwiner map );
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 );
# CHECKED 28/09/11 jdb
#############################################################################
#O IsomorphismPolarSpaces( <ps1>, <ps2> )
# returns (<> <> true)java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
##
InstallMethod( IsomorphismPolarSpaces,
"for two similar polar spaces",
, IsClassicalPolarSpace]
function( ps1, ps2 )
return return PG(r*t-1,Size)
end )
# CHECKED 28/09/11 jdb
################################################################################
#O BlownUpSubspaceOfProjectiveSpace( <B>, <subspace> )
# This operation is just like its namesake except that it
# hasno checks
InstallMethod( IsomorphismPolarSpacesNC,
a boolean" java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
[ IsClassicalPolarSpace, IsClassicalPolarSpace,[IsBasis ],
( ,ps2 computeintertwiner)
local form1, form2, f, map, c1, c2, change, invchange, hom, ty1, ty2, n,
coll1, coll2, gens1, gens2 coll1, coll2, gens1, gens2,java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 33
ty1 := if not pg1!.basefield(^) then
ty2 := PolarSpaceType( ps2 );
f := ps1!.basefield;
Error" are compatible!;
form1 := QuadraticForm( ps1 );
form2 := fi
else
form1 := SesquilinearForm( ps1 );
form2 : SesquilinearForm(ps2 ;
fi;
1: ( )
c2 := if subspacetype=then
java.lang.StringIndexOutOfBoundsException: Range [21, 8) out of bounds for length 97
#ConvertToMatrixRepNC(change, f);
invchange := Inverse mat2:BlownUpMatb,;
: functionxjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
return VectorSpaceToElement(ps2, ShallowCopy This wr. thecanonicalbasisofthefieldover subfield.
end;
inv :=
return VectorSpaceToElement(ps1, ShallowCopy(InstallMethod(BlownUpSubspaceOfProjectiveSpaceBySubfield,
end;
map := IsField, IsSubspaceOfProjectiveSpace
(subfieldsubspace)
java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 53
(The first is asubfieldof basefieldof theprojective!);
#map fromgens2togens1
twinerprefun :BasisAsVectorSpacesubfield))
local y;
y:= MutableCopyMat( !mat )
#return ProjElWithFrob( invchange * java.lang.StringIndexOutOfBoundsException: Range [0, 53) out of bounds for length 7
return ProjElWithFrob(invchange*x *change^y!frob^1,y! );
java.lang.StringIndexOutOfBoundsException: Range [37, 22) out of bounds for length 89
end;
# map from gens1 to gens2
twinerfunc := function( y )
localx;
x := for a basis and a subspace space,
#ProjElWithFrob *xy!frob, y.rob );
return java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
end;
if computeintertwiner then
if (HasIsCanonicalPolarSpace )IsCanonicalPolarSpaceps1) or
HasCollineationGroup(
coll1 := r=rt/;
gens1 := pg1:PG(1qt;
: Listg,twinerfunc;java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
coll2 := GroupWithGenerators( v1:List(1.r,-v[i1)t1.i*t]*;
hom :=GroupHomomorphismByFunctioncoll1,coll2,twinerfunc );
SetIntertwinerflag: =VectorSpaceToElementp,;
java.lang.StringIndexOutOfBoundsException: Range [0, 26) out of bounds for length 5
end)
HasCollineationGroup( ps2 ) then
coll2 := CollineationGroup( ps2 );
gens2:=GeneratorsOfGroup( coll2 );
gens1 := List(gens2, twinerprefun );
coll1GroupWithGeneratorsgens1;
hom: GroupHomomorphismByFunction(coll1,, , twinerprefun)
winer ,hom);
UseIsomorphismRelation(coll1,coll2);
elsejava.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
(,1, " intertwiner .One polarspaces must a group ";
fi;
fi;
return map;
end );
# CHECKED 28/09/11 jdb
################################
#O IsomorphismPolarSpacesNC(#basis the subspaceis blownup
#returnsIsomorphismPolarSpacesNC <ps1><,true )
##
smPolarSpacesNC
"for two similar flag:true;
[ java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
t=!.;
return K:=GF(q^t)
;
#############################
## Field reduction / subfield morphisms. Helper operations.
################################
28/911jdb
#########################################
#O flag=falsejava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
# returns the preimage of return flag;
##
InstallMethod( ShrinkMat,
"for a basis and a ##################################################
],
function( B, ########################################
## THERE WAS A BUG IN ## THERE WAS A BUG IN THIS
## NOT ALL#A reference projectivespaces is
#Here is theexplanation could go into thedocumentation:
# TheresultofBlownUpMatB) is the matrix, each entry= isreplacedby
## the dxd matrix M_a, ########################################################
# Kseen - vectorspace F.This means if
## B={b_1,b_2#< is aprojective a K,geom1>is a projectivespace
y we wanttoimplementShrinkMat we to the inputis
#that <geom1>and <geom2 have the vectorspace K this
canbe done asfollows. A admx matrix dxd block, say M,
## # K.
## (Since # Important note note: the intertwinerhas as * group* only
## is basicallythe java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
## "for two projec two projective spacesabasis"java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
unction pg1,, )
d:=Size(
ecs=();
m:=NrRows(mat);
n:=rColsmat)
# First f1 : pg1!.basefield;
if not (IsInt(m/d) and IsInt(/d)then
Error("The =Size(1;
fi;
blocks:=List(Cartesian([0..m/d-1], 2: Sizef)java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 12
iin[.m java.lang.StringIndexOutOfBoundsException: Range [21, 22) out of bounds for length 21
row=]
for j d2 := d1 : *tjava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
#submat:=blocks[(ifnot (sBasisbasis)andf1=GF((basis.)basis.d)and f2=F(asis!q *(asis.d)=d2
submat=(-)(/d+;
checklist:= i;
if Size(
Error fun :=function(subspace #Thismap up a subspace geom1 to subspaceof geom2
f;
#BlownUpSubspaceOfProjectiveSpacex;
od;
Add(newmat,row);
d;
return newmat;
DOWE NEED TOUSEConvertToMatrixRepNC java.lang.StringIndexOutOfBoundsException: Range [44, 45) out of bounds for length 44
end);
InstallMethod( ShrinkMat,
"for a field, a subfield and a matrix"return VectorSpaceToElement(pg2,mat2);
[IsField,sField,IsMatrix ],
function( f1,f2,mat )
gives the sameresult asShrinkMatBmat the natural basis returned by
# Basis(AsVectorSpacelocalflagb,mat1,,,,v1,i,ecsjava.lang.StringIndexOutOfBoundsException: Range [55, 56) out of bounds for length 55
local ;
B:=Basis(AsVectorSpace(f2,Error(Theinputisnotintherangeofthe map";
return fi;
end)
###########################################
#Obasvecs:BasisVectorsbasis)
# f2 is a subfield of f1 vecs := Unpack(subspace)java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
lengthd2t d2=(2,andt[1f]
# using the natural basisspan =VectorSpaceToElement(g2,lownUpMat(basis,at1)
##
nstallMethod ShrinkVec,
"for a field, a subfield and a vector",
[ IsField, IsField, java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 19
f2, )
local (Theinputisnotintherangeofthe field map";
t:=Dimension(AsVectorSpace(f2,f1));
d1:=Length(vfijava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
if IsInt(d1) then
java.lang.StringIndexOutOfBoundsException: Range [0, 9) out of bounds for length 7
return List([1..d1],i->v{[ElementsOfIncidenceStructure(pg2),
else
Error("The length of v is not divisible by fun,,prefun);
fi;
end );
###################Nowcreating intertwiner
#O # /14 checktoseem projectivity
#
InstallMethod( ShrinkVec,
"for a field, a subfield and a vector",
[ IsField, IsField, IsVector, IsBasis localimage;
function( f1,f2,v,basis )
localt,,d2basvecs;
t:=Dimensionreturn fail;
d1:Info(InfoFinInG,1 "<l>is not aprojectivity)java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
if IsInt(d1) then
basvecs:BasisVectorsbasis);
return List(image, f2;
else
Error("The length of v is not Error("The length of v is not
fi;
end );
#####################################
#O ( <>,<pg1>)
# blows up a projective space returnC(preimage,)
#
InstallMethod
"for a g1 =ProjectivityGroup( );
[ IsBasis, IsProjectiveSpace ],
functionbasis,java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
local q,t,r,pg2, g2:= Group()
q:=basis SetSize(g2,Size(1))
t:basis!.;
sefield=GF(q^t) then
Error("The basis and the projective space are not compatible!");
fi;
r:=Dimension(pg1)+1;
pg2:=PG(r*t-1,q);
pg2java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
end );
################################## <>isa field ,<eom1 a space
#( subfield, <pg> )
# blows up projective space geom2 thathasthe vectorspace K.This
#java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
InstallMethod( BlownUpProjectiveSpaceBySubfield,
"( NaturalEmbeddingByFieldReduction,java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
[ IsField, IsProjectiveSpace ],
function(subfield,pg)
t,,ield;
field:=java.lang.StringIndexOutOfBoundsException: Range [0, 31) out of bounds for length 14
if notreturn (,2,asis)
Error("The first argumentend )
fi;
=((ubfieldfield));
r:=Dimension(pg) NaturalEmbeddingByFieldReductiongeom1> geom2><> )
(subfield);
end ); a field extension vector over K, yields
# CHECKED 1/10/11 jdb
#######################################
#O BlownUpSubspaceOfProjectiveSpace( <B>, <subspace> )
# blows up a subspace of a projective space by field reduction
##
InstallMethod( "for two projective spaces and,
or a anda subspace a projective space",
IsBasis,IsSubspaceOfProjectiveSpacejava.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
function(basis,subspace)
local pg1,q,t,r,pg2,java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 0
pg1:=AmbientGeometry(subspace);
q:basis!q;
t:=basis!.d;#O (<> <geom2>)
if not# geom2> aprojectivespaceovera field,< isa projectivespace
Error("The basis and the subspace are not compatible!");
;
r:=Dimension(pg1)+1;
pg2:PG(*-,)java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
mat1=!obj;
if subspaceI(NaturalEmbeddingByFieldReduction
or two projective spaces"
fi;
mat2:=BlownUpMat(basis,mat1);
return VectorSpaceToElement( return VectorSpaceToElement(pg2,mat2);
end );
#######################################
#O java.lang.StringIndexOutOfBoundsException: Range [48, 46) out of bounds for length 72
# blows up a subspace of projective space by
# This is w.r.t. to the canonical basis
##
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 0
"for a field java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ java.lang.StringIndexOutOfBoundsException: Range [0, 10) out of bounds for length 0
function(subfield#######################################
local pg POLAR SPACES
pg:=#################################################
field:=LeftActingDomain(java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 0
subfield inSubfieldsfield)then
Error("# [1] from g point :" ofclassicalgroups byNickGill
fi;
basis:=# [2] from fi geometry pointof :"ield sets geomety"byLavrauw
BlownUpSubspaceOfProjectiveSpacesubspace)
end );
######################################java.lang.StringIndexOutOfBoundsException: Range [77, 78) out of bounds for length 77
#
# checks
##
InstallMethod( IsDesarguesianSpreadElement,java.lang.StringIndexOutOfBoundsException: Range [0, 44) out of bounds for length 1
"for a basis and a subspace of a projective space"java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
[ IsBasis, IsSubspaceOfProjectiveSpace ],
function(basis,subspace)
ocal,,,pg1,pg2,basvecs,v1,m,rtrjava.lang.StringIndexOutOfBoundsException: Range [52, 53) out of bounds for length 52
flag:=true;
q:basis!q
t:=basis!.d;
=(subspace;
rt:=Dimension(pg2)+1;
if not (Dimension(subspace)+1 = t and IsInt(rt/t)) then
flag:=false;#withbasis <asis> This operationthen
else
r:=
pg1:=PG(r-1,#REMARK ml 10/13)The argumentisa basisofL .T justgives
:BasisVectorsbasis)
java.lang.StringIndexOutOfBoundsException: Range [17, 4) out of bounds for length 44
v1=(1.],->{[i)*1.*]}basvecs;
mat=(,v1];
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 0
fi;
return flagjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
end );
# To be done: check this method! (but java.lang.StringIndexOutOfBoundsException: Range [0, 40) out of bounds for length 30
####################
#O IsBlownUpSubspaceOfProjectiveSpace( <Bl mat1f1d1t,V2,V1,hib2,,mat2,,row,,bil2
# a subspace blown upusingreduction w.t basis
##
InstallMethod( IsBlownUpSubspaceOfProjectiveSpace,
"for a basis and a subspace of a projective t:Dimension((,1)java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
[ IsBasis, IsSubspaceOfProjectiveSpace ],
# It's important that we java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 15
# since for java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 10
# basis with respect to which the subspace is blown up.
function(for j in [1.
flagqF,,,pg2,rtrpg1,,at1s,,,v1,;
flag:=true;
q:=basis!.q;
F:=GF(q);
t:=basis!.d;
K=GF^t;
pg2: bil2;
rt:=Dimension(pg2)+1;
if not (IsInt(rt/t) and IsInt((Dimension(subspace)+1)/t)) then
flag:=
else
:rt/;
pg1:=PG(rO BilinearFormFieldReduction(<bil1>, <f2>, <alpha> )
:(basis);
mat1:=[];
span:=[];
repeat
; untilnot xin span;
v:=Coordinates(x);
v1:=List([1..r],i->v{[(i-1)*t+1..i*t]}[IsBilinearForm IsField, IsFFE ,
Add(mat1,v1);
span:VectorSpaceToElementpg2,lownUpMatbasis,)java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
Dimensionspan)Dimension(subspace;
if not span = end )
flag:= false;
fi;
fi;
returnflag;
end );
#############################################################################
# PROJECTIVE SPACES
#############################################################################
#
# A recent reference on field reduction of projective spaces is
# "Field reduction and linear sets in finite geomety" by Lavrauw
and Voordepreprint 2013)
##############operation .
#O # There is also a version of this operation does use
# isa spaceover ,<eom1isaprojectivespace
# over a field extension L, and java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
# that <"for quadratic ",
# returnsthe ,i.withrelation the basis
# of L over K.
# Important note: the intertwiner has as source the *#f2 a ofbasefield thequadraticformq1
#thefullcollineationgroupjava.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
##
InstallMethod( NaturalEmbeddingByFieldReduction,
"for two projective spaces and a basis",
[ IsProjectiveSpaceif .java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
,, )
# morphism a prefuncwith built-ncheck.
local map, f1, d1, d2, t, fun, prefun, g1, gens, newgens, g2, twiner, homlocal flag,,,span,,,1i;
f1 := pg1!.java.lang.StringIndexOutOfBoundsException: Range [8, 1) out of bounds for length 19
q1 := Size(f1);
q2 := Size(f2);
t =LogInt(1,2;
if not q2^t = q1 then
Error( "<f2> is not a subfield of the base field of <pg1> ");
fi;
d1 := pg1!.dimension (subspace)+)t
d2 := d1*t;
pg2 := ProjectiveSpace( pg2 := ProjectiveSpace(d2
if not (IsBasis(basis) and f1=GF((basis!.q)^basis basvecs:BasisVectorsbasis;
Error("The basis is not a basis or is not compatible with the basefields of the geometries"span=]
f;
:function java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 92
mat1mat2;
#return BlownUpSubspaceOfProjectiveSpace(basis,x);
java.lang.StringIndexOutOfBoundsException: Range [16, 12) out of bounds for length 34
if subspace!.type = 1 then
mat1 := [subspace!.obj];
fi;
mat2java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 11
return VectorSpaceToElement( Error("The input is not in the the field!";
end;
prefun :return VectorSpaceToElement(ps1mat1)java.lang.StringIndexOutOfBoundsException: Range [46, 47) out of bounds for length 46
local flag,basvecs,mat1,span,x,v,v1,i,vecs ap=GeometryMorphismByFunction((ps1),ElementsOfIncidenceStructure),
true
if not subspace in pg2 then
Error("The input is not in the range of the field reduction map!");
fi;
((Dimension()1/)then
flag := false
else
basvecs:=BasisVectors(basis);
vecs(subspace);
mat1 := List(vecs,x->List([1..d1],i->x{[(i-1)*t+HasCollineationGroup ps1) then
span := VectorSpaceToElementhom:=function(m)
not = then
flag := false;
fi;
fijava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
if flag= false then
Error("The input is not in the range of thelocal preimage;
fi;
preimage :=ShrinkMatbasis,Unpackm!);
end;
return(preimage )java.lang.StringIndexOutOfBoundsException: Range [67, 68) out of bounds for length 67
torSpaceToElementpg1S(,x!objjava.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
#end;
map := GeometryMorphismByFunction(ElementsOfIncidenceStructure(pg1),
ElementsOfIncidenceStructure(pg2),
twiner: GroupHomomorphismByFunction(, hom,hominv;
(map true )
## Now creating intertwiner
## 12/6/14: added check to see
hom := function(m)
local image;
if not IsOne(m!.java.lang.StringIndexOutOfBoundsException: Range [0, 32) out of bounds for length 0
returnfail;
Info(InfoFinInG, 1, "<el> is not a projectivity");
;
ONaturalEmbeddingByFieldReduction geom1,f2><alpha,basis java.lang.StringIndexOutOfBoundsException: Range [71, 72) out of bounds for length 71
#ConvertToMatrixRepNC( image, f2 );
return CollineationOfProjectiveSpace(image, f2);
end;
hominv := function( m )
local preimagejava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
if not IsOne(m!.frob) then
return fail;
ty";
fi;
preimage :=end )java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
#second particular version: user wants a particular alpha, and bool, agrees with basis.
return CollineationOfProjectiveSpacepreimage );
end;
g1 := ProjectivityGroup( pg1 );
:GeneratorsOfGroup g1 ;
newgens := #where basis>is the canonical basis BaseFieldg)overf2>
g2 :#
SetSize(g2, Size(g1));
twiner := GroupHomomorphismByFunction(g1, g2, hom, hominv);
twiner;
return map;
end );
################### function(s1f2,,)
#O NaturalEmbeddingByFieldReduction NaturalEmbeddingByFieldReductionp,2,lphaBasisAsVectorSpace(2BaseFieldps1)),bool;
field field K,<geom1 java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 7
L andconsideringLas a space ,yields
# that a java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 7
# operation returns the natural embedding, i.e. with relation to the standard basis of
# L over K.
##
NaturalEmbeddingByFieldReduction
"for a projective space and ()then
[ IsProjectiveSpace IsField,
function( pg1, f2 )
local basis;
basis:=asis(sVectorSpace(pg1!);
return NaturalEmbeddingByFieldReduction(pg1,f2,basis);
end );
########################
#O NaturalEmbeddingByFieldReduction type1 hermitian andtype2="elliptic IsEvenInt()and IsOddInt()
# <geom2> is java.lang.StringIndexOutOfBoundsException: Range [2, 1) out of bounds for length 24
a extension and Lasaspaceover,
# that <geom1> and <geom2> have the same java.lang.StringIndexOutOfBoundsException: Range [0, 48) out of bounds for length 21
# operation returns the natural embedding, i.e. form2: (form1,f2,alpha,basis);
# of L
##
InstallMethod( NaturalEmbeddingByFieldReduction,
tive spaces and a "java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 51
is )
return NaturalEmbeddingByFieldReduction #ap :=NaturalEmbeddingByFieldReduction,f2,alpha java.lang.StringIndexOutOfBoundsException: Range [67, 66) out of bounds for length 69
end);
# CHECKED 28/09/11 jdb
############################################################################ fi;
#O NaturalEmbeddingByFieldReduction( <else#form1isaquadraticform
# <geom2 iftype1="yperbolic"type2="yperbolic" then
# over a field extension L, and considering L as a vector space over Info(InfoFinInG, 1, "These polar spaces are suitable for field redu 1,Thesespaces java.lang.StringIndexOutOfBoundsException: Range [56, 55) out of bounds for length 78
#that<geom1>and <geom2 have thesame ambient vectorspace over K, then this
# operation returns the natural embedding, i.e. with relation to the standard basis of
# L over K.
##
InstallMethod ,
"for two projective spaces",
[ java.lang.StringIndexOutOfBoundsException: Range [39, 20) out of bounds for length 42
function( pg1, pg2 )
local basis
basis:=Basis(AsVectorSpace(pg2!.basefield form2:=QuadraticFormFieldReduction(orm1,f2,alpha,asis);
return NaturalEmbeddingByFieldReduction(pg1,pg2!.basefield,basis);
end );
######################### alpha : One(1;
# POLAR #map : NaturalEmbeddingByFieldReduction( ps1, 2,alpha,basis )
################form2 :=,2alpha,basis;
#
# Two references on field that will use for the embedding
# # dependson the typeof . isometrytype is determined
# (New Zealand J. Math gamma being a or,where
# [2] from # (-1)^(r-1)/2 the determinant the bilinearform .
# and Van de Voorde (preprint 2013)
#
# We will use [2] as it is more convenient to us.
####
# #somewhat easier to work with than the conditions in Gill's paper.
# java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0
#
#####
#############################################################################
#O bil1=AssociatedBilinearForm(form1); # important to use this instead of
nonsquare
# <alpha> determines the linear map from f1 to <f2>, f1 the field extension of <f2>
#with basis basis>. This operation then
# returns the bilinear form L(bil1(.,.)), L(x) = T(alpha*x), T the trace from <f1> to <f2>.
##
# REMARK (ml 31/10/13) The fourth argument is a basis of L over K. This alpha := First(f1, a >notIsZeroais_square *,f1) )java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
# extra functionality to the user, and it is used as third argument in java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 71
#
# There is also elif type1="arabolic"java.lang.StringIndexOutOfBoundsException: Index 101 out of bounds for length 101
# a basis as fourth argument. In this case the standardgamma=(-)(r1)2) (!matrix/;
InstallMethod( BilinearFormFieldReduction,
"for a bilinear form and a field",
[ IsBilinearForm IsFFE ,
function(bil1,f2,alpha,basis)
# f2 is a subfield of the basefieldmap:NaturalEmbeddingByFieldReduction,f,alpha,basis ;
localform2: QuadraticFormFieldReductionform1falpha,basis);
mat1:=bil1!.matrix;
f1
d1:=NrRows(mat1);
t:=Dimension(AsVectorSpace(f2,f1 Error(These polar nsuitablefield.;
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) : !.bj;
od;
Add(mat2,row);
od;
bil2:=BilinearFormByMatrix(mat2,f2)java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
return bil2;
end );
###################################################end;
#O BilinearFormFieldReduction( <bil1>, <f2>, <alpha> )
# The same as above, #this version of prefun new(//.
#flag=;
InstallMethod( java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
"for a bilinear form and a field",
[ IsBilinearForm, IsField, IsFFE ],
functionbil1,java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
# f2 is a subfield of the basefield of the bilinear form bil1
return BilinearFormFieldReduction(bil1,f2,alpha,Basis(AsVectorSpace(f2,bil1# ps2),
nd);
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
#O QuadraticFormFieldReduction( <qf1>, <f2>, <alpha>, <basis> )
# <bil1> is a bilinear form over <f1>, <f2> is a subfield of <f1>. This operation
# returns the SetIsInjective( map, )
##
# REMARK (ml 31/10/13) The fourth argument is a if H( ps1)andIsCanonicalPolarSpace ps1 )ps1 ) or
# 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 notlocal image;
# a basis as fourth argument. In this case the standard GAP basis is used.
InstallMethod,
"for a quadraticreturn (cps2*image cps2inv,f2)java.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81
[ IsQuadraticForm, IsField, IsFFEhominv=function )
function(qf1,f2,alpha,basis)
# f2 is a subfield of the basefield for the quadratic form q1
local f1,basvecs,d1,t,d2,V2,V1,phi,b2,b2vecs,mat2,i,bil1,j,qf2;
f1:=qf1!.basefield;
basvecs:=BasisVectors(basis);
:NrRows(qf1!.)java.lang.StringIndexOutOfBoundsException: Range [25, 26) out of bounds for length 25
t:=Dimension(AsVectorSpace(f2,f1));
d2:=d1*t;
V2:=f2^d2;
V1:=f1^d1;
b2:=Basis(V2);
b2vecs:=gens ( );
mat2:=IdentityMat(d2,f2);
for i in[..d2] java.lang.StringIndexOutOfBoundsException: Range [20, 21) out of bounds for length 20
mat2[i,i]:=Trace(f1,f2,alpha*((ShrinkVec(f1,f2,b2vecs[i],basis))^qf1))SetSizeg2 (1)
;
for i in [1..d2-1] do
for j in [i+1..d2] do
mat2[i,j]:=Trace(f1fi;
-(ShrinkVec(f1,f2,b2vecs[i],basis))^qf1-(ShrinkVec(f1,f2,b2vecs[j],basis))^qf1));
#mat2[j,SetIntertwinerm,hom)java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
od;
od;
:QuadraticFormByMatrixmat2,)
return qf2;
end );
##################################################################java.lang.StringIndexOutOfBoundsException: Index 85 out of bounds for length 85
# #O QuadraticFormFieldReduction( <qf1>, <f2>, <alpha> #This takestwopolar spacesand embedding
# The obtained by field reduction if such embedding exists, it returnsanerrorjava.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 89
##
InstallMethod( QuadraticFormFieldReduction,
"for a quadratic form and a field",
[ IsQuadraticForm, IsField, IsFFE ],
function(qf1,f2,alpha)
local basis,qf2;
# f2 return NaturalEmbeddingByFieldReductionps1,s2t)
:Basis(AsVectorSpace(2,!b);
qf2:=QuadraticFormFieldReduction(qf1,f2,alpha,basis);
return qf2;
end );
########################################################################
#O HermitianFormFieldReduction( <hf1>, <f2>, <alpha>, <basis> )
# <hf1> is a java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
# returns the form T(alpha*bil1(.,.)). java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 1
# yieldseither abilinear or hermitian formjava.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
##
# REMARK (ml 31/10/13) The fourth argument is a basis of # CHECKED/0614
theuser andit is as inthe
# operation ShrinkVec.
# There is also#########################################
# a basis as fourth argument. In this case the <geom2>, two projective of
MethodHermitianFormFieldReductionjava.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
",
[ IsHermitianForm, IsField, IsFFE, IsBasis ]" two projectivespaces",
hf1,,,)
# f2 is
sasquare
local f1,d1,t,d2,V2,V1,phi,b2Error(Dimensionsandorfieldareincompatible) return;
f1:=hf1!.basefield;
d1:=NrRows(hf1!.matrix);
t:=Dimension(AsVectorSpace(f2,f1));
d2:=d1*t;
java.lang.StringIndexOutOfBoundsException: Range [51, 3) out of bounds for length 51
V1:=f1^d1;
b2:=Basis(V2);
b2vecs:=BasisVectors(b2);
mat2:=[];
for i in [1..d2] do
r:;
in1.2 java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
Add(row,Trace(f1,f2,alpha* (,hom;
od;
Add(mat2, map, ,q1,q2 ,,g1 , ,gens2 ,
od
# checking parity ,basis d , bmat ,, c1 c2,form2;
if IsOddInt(t) then
hf2:=HermitianFormByMatrix(mat2,f2);
else
hf2#
fi;
return hf2;
end );
########################Allpossible cases:
# #O HermitianFormFieldReduction(
# The #W- (uadratic)
##
InstallMethod( HermitianFormFieldReduction,
"for a hermitian form and a field",
[,IsField,IsFFE ],
function(hf1,f2,alpha)
# f2 is a subfield of the basefield for theif geom1!. = geom2!.dimensionand IsSubsetf,)thenjava.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
the basefieldis always a
local basis;
basis:=Basis(AsVectorSpace(f2,hf1!.basefield));
return HermitianFormFieldReduction(hf1,f2,alpha,basis);
)
##########################################################
#
# 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 form := QuadraticFormByMatrixgram f2)
#
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
d2 := d1*t;
#1. the polar
if type1 in ["hyperbolic","elliptic","parabolic"] then
qf1:=QuadraticForm(ps1);
=,2alphabasis;
q)
Error("The field reduction does not yield a natural;
else ConvertToVectorRepNC(nvecf); #,also if appearantly vec is already f1
fi;
# 2. the polar space is of symplectic : BlownUpMat(basis,vec)
: SumIntersectionMatb,bmat[] #hard coded operation:)
bil1:=SesquilinearForm(ps1);
bil2:=nvec := List(list->(,,x);
ps2:=
# 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 )
,java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 30
if subspace!.type = 1 then
mat1 := [subspace!.obj];
fi;
: (basism)
return VectorSpaceToElement(ps2,mat2);
end;
#
# local ymat := *Unpackymat)* change here!
# projprefun:=emymat = ymatFirsty[1]-notIsZero());
# return if not ForAll( Flat) ->i
# new version (2/7/14, much faster since we avoid the repeat loops and Random ;
prefun := function( subspace ) # This map is the inverse of func
if ( geom1 )then
flag:=true;
if not subspace in ps2 then
Error("The input is not in the range of the field reduction map!");
fi;
if not IsInt((Dimension(subspace)+1)/t) then
flag:=false;
basvecs:=BasisVectors(
mat1:=[];
span:=[];
vecs :=# CHECKED 28/9/11jdb
######################################################java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
=VectorSpaceToElement(),(asismat1);
#the returns NaturalEmbeddingBySubfield( <geom1>, <geom2>, true )
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 [IsClassicalPolarSpace ]
;
return VectorSpaceToElement(ps1return NaturalEmbeddingBySubfield(geom1,geom2,true)
end;
map := GeometryMorphismByFunction(ElementsOfIncidenceStructure(ps1), ElementsOfIncidenceStructure(ps2),
fun, false, prefun########################################
SetIsInjective( map, true );
## Now creating java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 1
if computeintertwiner then
if ( if (HasIsCanonicalPolarSpace(
HasCollineationGroup( ps1 ) then
hom := function( m )
local image;
image := BlownUpMat(basis, m!.mat);
ConvertToMatrixRepNC( image, f2 );
return CollineationOfProjectiveSpace(image, f2);
end;
java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 35
local preimage;
preimage := ShrinkMat(basis, Unpack(m!.mat));
ConvertToMatrixRepNC( preimage, f1 );
return CollineationOfProjectiveSpace(preimage, f1);
g1 := IsometryGroup( ps1 );
gens := GeneratorsOfGroup( g1 );
newgens: (,hom;
g2 := Group( newgens );
SetSize(g2, Size(g1));
twiner(g1, g2,hom )java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71
SetIntertwiner( map,Wvectors,mb,compl , ,
java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
fi;
return map;
end );
first:userwantsparticular ,andbasis, booltrue.
#############################################################################
#O NaturalEmbeddingByFieldReduction( <geom1>, <f2>, <alpha>, <basis> )
# returns NaturalEmbeddingByFieldReduction( <geom1>, <f2>, <alpha>, <basis>, true )
#
InstallMethod (NaturalEmbeddingByFieldReduction,
"for a polar space and a field, a finite field element, and a basis",
[IsClassicalPolarSpace, IsField, IsFFE, IsBasis],
function(ps1,f2,alpha,basis)
return NaturalEmbeddingByFieldReduction(ps1,f2,alpha,basis,true);
);
#second particular version: user wants a particular alpha, and bool, agrees with java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 52
### NaturalHomomorphismBySubspace code
#O NaturalEmbeddingByFieldReduction( <geom1>, <f2>, <alpha>, java.lang.StringIndexOutOfBoundsException: Range [3, 4) out of bounds for length 3
# returns NaturalEmbeddingByFieldReduction( <geom1>, <f2>, <alphamb: MutableBasis( f )
#compl=[;
#
InstallMethod (NaturalEmbeddingByFieldReduction,
"for a polar space and a field, a finite field element, and a boolean",
[IsClassicalPolarSpace, IsField, IsFFE, IsBool],
function(ps1,f2,alpha,bool)
returnon(ps1,,,(AsVectorSpacefBaseField))bool;
end );
#third particular version: user wants a particular alpha, agrees with basis and bool.
#######################################################################java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
#O NaturalEmbeddingByFieldReduction( <geom1>, <f2>, <alpha> )
NaturalEmbeddingByFieldReductiongeom1>, f2,<alpha>, basis> true)
# where <basis> is the canonical basis of BaseField(geom1) over <f2#Matrixofthelinear we
#
(NaturalEmbeddingByFieldReduction
"for a polar space, a field, and a BasisVectors( canbas ));
[IsClassicalPolarSpace, IsField, IsFFE],
function(ps1,f2,alpha)
return NaturalEmbeddingByFieldReduction(ps1,f2,alpha,Basis(AsVectorSpace(f2,BaseField( local;java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
;
# java.lang.StringIndexOutOfBoundsException: Range [31, 4) out of bounds for length 31
########################################## # !
#O NaturalEmbeddingByFieldReduction( <geom1>, <f2>, <bool> )
# returns NaturalEmbeddingByFieldReductionjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
# where <basis> is the canonical basis of BaseFieldy:= y[1];
# <one> is #onvertToVectorRep(,f;
# want the java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 6
#
InstallMethod (NaturalEmbeddingByFieldReduction,
"for a polar space, a field, and a re = java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
[x :=(.) c
functionif!. = x:x;fi
return NaturalEmbeddingByFieldReduction(ps1,f2,java.lang.StringIndexOutOfBoundsException: Range [0, 52) out of bounds for length 47
end );
# fifth particular version: user agrees with everything
#####################################
#O NaturalEmbeddingByFieldReduction( <geom1>, <f2ElementsOfIncidenceStructure(ps2), func, pre );
# returns NaturalEmbeddingByFieldReduction( <geom1>, <f2>, <alpha>, <basis>, true )
# where <basis> is the canonical basis of BaseField(geom1) over <f2>, and
# <alpha> is One(f2
#CHECKED 28// jdb
"for a polar space and a field",
I, IsField,
function(ps1,f2)
return NaturalEmbeddingByFieldReduction(ps1,f2,One(f2),Basis(AsVectorSpace(f2,BaseField(ps1))),true# Thisoperation just its namesake that itjava.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
end );
# CHECKED 28/09/11 jdb
###############################################"for projective space and a singular subspace",
#O NaturalEmbeddingByFieldReduction( <geom1>, <geom2> )
# returns NaturalEmbeddingByFieldReduction(<geom1>, <geom2> )
##
InstallMethod( NaturalEmbeddingByFieldReduction,
[ IsClassicalPolarSpace, IsClassicalPolarSpace ],
function(geom1 geom2 )
return NaturalEmbeddingByFieldReduction( geom1, geom2, true );
end );
# ml 31/10/# ml 31/10/2013,mb ,gen,bas,img
# newprefun2//jdb
#################################
# O NaturalEmbeddingByFieldReduction
# This NaturalEmbeddingByFieldReduction takes two polar spaces and returns an embedding
# obtained by field reduction if such an embedding existsb: !o;
##
InstallMethod (NaturalEmbeddingByFieldReduction,
"for two classical polar spaces",
[IsClassicalPolarSpace, IsClassicalPolarSpace, IsBool],
function(ps1, ps2, computeintertwiner)
#####################################################################
# List of possible embeddings of ps1 in# if vhas, geometryjava.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
#
# Hermitian
H- H (todd)
# H -> W (t even)
# H -> Q+ (t even, r even# This was adapted fromThomas Breuer's
# H -> Q- uralHomomorphismBySubspace code
#
# Symplectic
# W -> W (always)
#
# Orthogonal
# r even
+ ->Q a)
# Q- -> Q- (always)
# r odd
# Q -> Q (t odd, q odd)
# Q -> Q+ (t even, q odd)
# Q -> Q- (t even, q odd)
#####################################################################
type2form1 ,,newps,sigma map,
gamma, bil1, q, n, alpha, basis, is_square, iso, form2, fun, prefun,
c1, c2, cps2, cps2inv #Compute linearmapping by images.
type1: (ps1;
type2 := PolarSpaceType(ps2);
if type1 in ["hyperbolic", "parabolic", "elliptic"] then
form1 := QuadraticForm(ps1);
else
form1 := SesquilinearForm(ps1);
fi
d1 := ( canbas))
f1 := ps1!.basefield;
f2 := ps2!.basefield;
q := Size(f2);
if not IsSubset(f1, y := Lis : (.,->Coefficients,)*;
Error("Fields are y : SemiEchelonMat(y!vectors;java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
fi;
t := Log(Size(f1),q);
r := ProjectiveDimension(ps1)+1;
basis := ConvertToVectorRepy, f);
is_square := function(x, f) return IsZero(x) or IsEvenInt(LogFFEelse
if IsSesquilinearForm()then
if type1 = "hermitian" and type2 = "hermitian" java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 17
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
Info(InfoFinInG, 1, y)
if IsEvenInt(q) then
# need alpha to be fixed by sigma:
alpha := One(f1);
else
# need sigma(alpha)=-alpha
sigma := Sqrt(Size(f1));
alpha :ConvertToMatrixRepNC(x,);
fi;
#map := NaturalEmbeddingByFieldReduction( ps1, f2, alpha, basis );
form2 := HermitianFormFieldReduction(form1,f2,alpha,basis);
elif type1 = "hermitian" and type2 = "hyperbolic" and IsEvenInt(t) and IsEvenInt(r)
and IsOddInt(q) then
Info(################################################
alpha := One(f1)O / <s> v )
#map := NaturalEmbeddingByFieldReduction( ps1, f2, alpha, basis );
: HermitianFormFieldReduction(form1,f2,alpha,basis);
elif type1 = "hermitiansubspace of <ps>
and #
Info(InfoFinInG, 1, "These polar spaces are suitable for field java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 22
alpha := One(f1);
# : NaturalEmbeddingByFieldReduction(ps1,f2,alpha,basis )
form2 := HermitianFormFieldReduction(form1,f2,alpha function(ps,v java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
elifSubspacedoesnotbelongthe
Info(InfoFinInG, 1, "These polar spaces are suitable fi;
alpha (NaturalProjectionBySubspace ps,v !geometryjava.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
#map := NaturalEmbeddingByFieldReduction( ps1, f2, alpha, basis );
form2 := BilinearFormFieldReduction(form1,f2,alpha,basis);
Error("These polar spaces are not java.lang.StringIndexOutOfBoundsException: Range [0, 46) out of bounds for length 20
fi;
#form1isa form
if type1 = "hyperbolic" and type2 = "hyperbolic" then
Info(InfoFinInG, 1, "These polar spaces are suitable for field reduction");
alpha := One(f1);
#map := NaturalEmbeddingByFieldReduction( ps1, f2, alpha, basis );
form2 := QuadraticFormFieldReduction(form1,f2,alpha,basis);
java.lang.StringIndexOutOfBoundsException: Range [24, 6) out of bounds for length 53
Info(InfoFinInG, 1, "hesepolar spacesare suitable field reduction";
alpha := One(f1);
#map := NaturalEmbeddingByFieldReduction( ps1, f2, alpha, basis );
form2 := ## map from geometry of incidentwith v
# tothe polar pace thesame butoflesser
Info(InfoFinInG, 1, "These polar spaces are suitable for field java.lang.StringIndexOutOfBoundsException: Range [0, 75) out of bounds for length 48
alpha := One(f1);
#map := NaturalEmbeddingByFieldReduction( ps1, f2, alpha, basis );
form2 := QuadraticFormFieldReduction(form1,f2,alpha,basis);
# If ps1 is Error(v is ";
# then the choice of alpha that we will use for the embedding
# depends on the isometry type of ps1. The isometry type is determined by
# gamma being a square or not, where gamma psdim =ps.;
# (-1)^(r-1)/2 f: !basefield
# b :Unpack(v!) cmat necessary infra
ion and linear sets finite" preprint] they are
# somewhat easier to work withif vdim =1 thenb=b;fi;
elif type1 = "parabolic" and type2 = "hyperbolic" and IsEvenInt(t) and IsOddInt(q) then
Info(InfoFinInG, 1, "These polar spaces are suitable for field reduction: Parabolic -> Hyperbolic");
bil1:=; java.lang.StringIndexOutOfBoundsException: Range [52, 51) out of bounds for length 75
# BilinearFormByQuadraticForm (see java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 0
gamma:=(m:M java.lang.StringIndexOutOfBoundsException: Range [35, 34) out of bounds for length 37
if q^(t/2) mod 4 = 1 then # the product for gen in BasisVectors( Basis( vs ) ) do
: First(f1,a>not IsZero()andnotis_square( a*amma,f1 ))
else # the product of alpha andAdd compl gen)java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
alpha := First(f1, a -> not IsZero(a) and is_square( a*gamma, f1 ) );
fi;
#map := NaturalEmbeddingByFieldReduction( ps1, f2, alpha, basis );
form2 := QuadraticFormFieldReduction(form1,f2,alpha,basis);
liftype1 ="" ="lliptic" andIsEvenIntt IsOddIntq then
Info(InfoFinInG, 1, "These polar spaces areimg= ( ,Length( compl );
bil1:=AssociatedBilinearForm(form1); # important to use this instead of
# BilinearFormByQuadraticForm (see documentation of the Forms package).
gamma:=((-1)^((r-1)/2)) * Determinant(basimgs:= Concatenation( List( Wvectors- zero)
if q^t/)mod = 1then #the product of alpha andgamma must bea square
alpha := First(f1, a -> not IsZero(a) and is_square( a*gamma, f1#onvertToMatrixRepbasimgs ) useles now.
else#the productof andgammamustbe anonsquare
alpha := First(f1, a -> not IsZero(a) java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 0
fi;
#map := NaturalEmbeddingByFieldReduction( ps1, f2, alpha, basis );
: (form1f2,lpha,basis)
else
Error( spacesarenotsuitable field .";
fi;
fi;
oCanonical )
if not IsCanonicalPolarSpace(ps2(a*m* TransposedMat(CompanionAutomorphism(ps)), f);
if type2 in ["parabolic", "hyperbolic", "elliptic"] then
c2 := BaseChangeToCanonicalelse
else
c2 := BaseChangeToCanonicalSesquilinearForm(s2)java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
fi;
cps2 := c2^-1 * c1;
else
cps2 := c1;
fi;
cps2inv := cps2^-1;
localy;
local mat1,mat2;
mat1 := subspace!.obj;
subspace!type =1then
mat1 := [subspace!.obj];
fi;
java.lang.StringIndexOutOfBoundsException: Range [14, 1) out of bounds for length 33
(Subspaceisnot an ofpolar")
end;
#this version v =then return
prefun := function( subspace ) # This map is the inverse of func and returns an error, or a subspace of geom1
local : List(npack(!.)i Coefficients(as,)basimgs #matunpackhere
flag:=true;
if not subspace in ps2 then
"he is not the of field reduction !)java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
;
if not IsInt((Dimension(subspace)+1)/t) then
flag:=false;
else
basvecs:=BasisVectors(basis);
vecs : fi;
mat1 := List(vecs,x->List([1..d1],i->x{[(i-1)*t+1..i*t]}*basvecs));
span := VectorSpaceToElement(AmbientSpace(ps2)fi;
if not span = subspace then
fi;
fi;
if flag = false then
(is map)
fi;
return VectorSpaceToElement(ps1,mat1);
end;
#iso := IsomorphismPolarSpacesNC(AmbientGeometry(Range(mapElementsOfIncidenceStructure(ps2), func, false, pre );
#morphism := GeometryMorphismByFunction(ElementsOfIncidenceStructure(ps1),
# ElementsOfIncidenceStructure(ps2),
# x -> iso!.fun(map!.fun(x)), java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 8
map := GeometryMorphismByFunction(ElementsOfIncidenceStructure(ps1),
cturep)
fun, false, prefun);
SetIsInjective( map, true##Thisoperationis its namesake except itjava.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
if (HasIsCanonicalPolarSpace( ps1 ) and IsCanonicalPolarSpace( ps1 )) or
( ) then
if computeintertwiner then
hom := function( m )
local image;
image [ IsClassicalPolarSpace, IsSubspaceOfClassicalPolarSpace ],
ConvertToMatrixRepNC( image, f2 );
return CollineationOfProjectiveSpace(cps2 * image * cps2inv, f2);
end;
hominv := function( m )
local preimage;
preimage : ShrinkMat(basis,cps2inv *Unpack(!.at)* cps2)
ConvertToMatrixRepNC( preimage, f1 );
return CollineationOfProjectiveSpace pstype, psdim,b,vdim, vs,func,,
end;
g1 := IsometryGroup( ps1 );
:=GeneratorsOfGroup g1)
newgens := List(
g2 : pstype :=PolarSpaceType)
SetSize(g2, java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
twiner := GroupHomomorphismByFunction(g1, g2, hom, hominv);
SetIntertwiner( map, twiner);
fi;
fi;
SetIntertwiner(map, hom );
return map;
end );
# added 2/7/14 jdb
#####################vs :VectorSpace(,perp(!obj)
# O NaturalEmbeddingByFieldReduction
# This NaturalEmbeddingByFieldReduction takes two polar spaces and returns ##ifvhas dimension j, then the new geometry
# obtained by field reduction if such an embedding exists, otherwise it returns an error.
##
InstallMethod (java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 0
"for two classical polar spaces",
[IsClassicalPolarSpace, IsClassicalPolarSpace],
function(ps1, ps2)
NaturalEmbeddingByFieldReductionps1p,truejava.lang.StringIndexOutOfBoundsException: Range [62, 63) out of bounds for length 62
end );
#################################################
#
#
# Embeddings by SUBFIELDS
#
#
#############################################################################
#y().ectors
# found another cvec/cmat bug if!type vdim =1then
####java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
eddingBySubfieldgeom1,geom2 )
# returns the embedding of <geom1> in <geom2>, two projective spaces of the
# same dimension, over the fields K and L java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 0
# of K.
##
java.lang.StringIndexOutOfBoundsException: Range [14, 1) out of bounds for length 27
"for two projective spaces",
eSpace]
function( geom1, geom2 )
local map, f1, f2, func, prefun, g1, g2,
gens1, gens2, hom, twinerfunc, twinerprefun;
f1 := geom1!.basefield;
f2 := map := Ge :=GeometryMorphismByFunction((ps,java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
if geom1!.dimension <> geom2!.dimension or not IsSubset(f2,f1) then
Error("Dimensions and/or field sizes are incompatible"); return;
fi;
func := x -> VectorSpaceToElement(geom2, Unpack(x!.obj)); #here an unpack is needed to avoid field incompatibilites.
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;
=Unpackx.)java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
if x!.type = 1 then
ConvertToVectorRep(xmat);
ConvertToMatrixRep(xmat);
fi;
return (geom1 xmat ; # he same here.
end;
map: ( ElementsOfIncidenceStructure),
ElementsOfIncidenceStructureifl. > orDimensionl. >3then
func, false, prefun );
SetIsInjective( map, true );
## Intertwiner...
twinerfunc := function( x )
calxmat;
if not IsOne(x!.frob) then
Info(InfoFinInG, 1, "<el> is not in the source java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 16
return fail;
;
xmat := x!.mat;
return CollineationOfProjectiveSpace( Unpack(x!.mat), f2 ); #here was an unpack needed!
end;
twinerprefun := function( y )
local
if not IsOne(y!.frob) then
Info(InfoFinInG, 1, "#CHECKED 28//
return fail;
fi;
:Unpacky.) h too
q :O ( <>)
if not q =# returns a of vectorsspanning line coordinates <java.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 80
(InfoFinInG,1,"el>is the intertwiner";
return fail;
fi;
return cmat note: l will be an ordinary matrix
end;
g1 := ProjectivityGroup( geom1 );
gens1 := GeneratorsOfGroup( g1 );
gens2 := List( [ ,
g2 := Group(gens2);
SetSize(g2, Size(g1));
hom := GroupHomomorphismByFunction(g1, g2, twinerfunc, twinerprefun);
SetIntertwiner(map, hom);
;
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 :-)
################################################################### +5, eachrow is uptoa.It out
#O needed factors arealways hesame:
# returns the embedding of <geom1> in <geom2>, [1,1,1,1,1,-1]. This explains the '-' at newmat
dimension Kand respectively whereLisanextension
## and latin do induce projectivity (,)java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
# The prefun principle is taken from NaturalDualityHermitian.
##
InstallMethod( NaturalEmbeddingBySubfield,
"for two polar spaces and "for two polar spaces and /atins are done mapping
[ IsClassicalPolarSpace, IsClassicalPolarSpace, IsBool ],
function(geom1,geom2, computeintertwiner
local map, f1, f2, q1 theimage is of Q+(5,q)
r, type1, type2,# swapsthegreeks/ of (,)
change, invchange, basis, d, n, bmat, one, i, c1, c2,#
#
# Check Kleidman and Liebeck Table 4.5.A
# There, the conditions on the polarity types can be found.
#
possible :
# W -> W
# W -> H (quadratic)
# O^eps -> H (q odd, quadratic)
# H -> H (odd degree)
one := nef;
#
f2 := geom2!.basefield; q2 := Size(f2);
type1 := PolarSpaceType(geom1);
type2 := PolarSpaceType(geom2);
form2 := SesquilinearForm( geom2 ); #will be changed in last case only
if geom1!.dimension = geom2!.dimension and IsSubset(f2, f1) then
r := LogInt(q2, q1);
gram := GramMatrix( SesquilinearForm(geom1) );
[ype1, type2]=[symplectic,h" r=2then
gamma := First( f2, t -> not IsZero(t) and IsZero(t^q1+ ps = (form)
newgram pg :=ProjectiveSpace3f;
form := HermitianFormByMatrix(newgram, f2);
elif [type1, type2] = ["symplectic" plucker:
form function(l java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
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, f2);
elif (IsOddInt(r) and type2 in ["elliptic", "parabolic", "hyperbolic"] and
type1 = type2) or (java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 21
[""",[parabolic","parabolic"]]
gram := GramMatrix( QuadraticForm(geom1) );return(pg )
form := QuadraticFormByMatrix(gram, f2
form2 := QuadraticForm(geom2);
else
Error"ot for these geometries\";
fi;
else
Error("Dimensions and/or field sizes are incompatible"); return;
fi;
:BaseChangeToCanonical ;
if not IsCanonicalPolarSpace( geom2 ) then
c2 := BaseChangeToCanonical( form2 );
change := c2^-1 * c1;
else
change := c1;
fi;
invchange := change^-1;
d := geom1!. newmat[4] := PluckerCoo(mat[3]mat[2]];
basis := Basis(AsVectorSpace(f1,f2));
n : Length(basis)
bmat=[e,id1][i[]]y>());
one := One(f1);
for i in [1..d] do
bmat[i,1+(java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
od;
func := function( el )
return VectorSpaceToElement(geom2,Unpack(el!.obj) * invchange);
end;
prefun := function( el )
local vec,nvec,list,bgen;
vec :=
if el!.type = 1 then
nvec := vec / First(vec,x->not IsZero(x));
if not ForAll( nvec, i -> i in f1 ) then
"isnot the the geometry morphism");
fi;
ConvertToVectorRepNC(nvec,f1); #necessary, also if appearantly vec is java.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 36
else
bgen := BlownUpMat(basis,vec);
list := SumIntersectionMat(bgen, bmat)[2]; #hard coded Meet operation :-)
nvec := List(list,x->ShrinkVec(f2,f1,x));
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 VectorSpaceToElement(geom1,nvec);
end;
map := GeometryMorphismByFunction( ElementsOfIncidenceStructure(geom1),
ElementsOfIncidenceStructure(geom2),
func, false, prefun );
SetIsInjective( map, true );
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( change * Unpack(x!.mat) * invchange, 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;
ymat := invchange * Unpack(y!.mat) * change; #here too!
ymat := ymat/First(ymat[1],x->java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 2
if ForAll(Flat(ymat) -> f1 )then
Info(InfoFinInG, 1, "<el> is not in the range of the java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 58
,ie,cs, iept ;
else
ConvertToMatrixRepNC(ymat,f1);
return CollineationOfProjectiveSpace(ymat,f1);
fi;
end
if HasCollineationGroup(geom1 ) then
g1 := SimilarityGroup( geom1 );
=GeneratorsOfGroup(g1 )
gens2 := List(gens1, twinerfunc );
g2 := Group(gens2);
( Size();
hom := GroupHomomorphismByFunction(g1, g2, twinerfunc, twinerprefun);
SetIntertwiner(map, hom);
else
Info ipts =List(ts,>mat;
fi;
return map;
end );
# CHECKED 28/09/11 jdb
########################################## fi;
#O NaturalEmbeddingBySubfield( <geom1>, <geom2> )
# returns NaturalEmbeddingBySubfield( <geom1>, <geom2>, true )
=ieptxInversePluckerCoordinates));
InstallMethod( NaturalEmbeddingBySubfield,
"for two polar spaces",
[IsClassicalPolarSpace,IsClassicalPolarSpace]
function( geom1, geom2 )
return NaturalEmbeddingBySubfield( geom1, geom2, true );
end );
#############################################################################
#
# PROJECTIONS
#
#############################################################################
# CHECKED 28/09/11 jdb
#############################################################################
#O NaturalProjectionBySubspace( <ps>, <v> )
# themorphism theprojectivespace the space of the
# subspace <v>. It is checked if <v> is a moment I recycle the interwiner from the appropriate'java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
InstallMethod( NaturalProjectionBySubspace,
"for a projective twinerprefun, coll1, g ,;
[ IsProjectiveSpace, IsSubspaceOfProjectiveSpace ],
function(ps, v)
## maphe formfor the resultingPluckercoordinatesis
## to the projective space java.lang.StringIndexOutOfBoundsException: Range [0, 31) out of bounds for length 25
local psdim, mat := NullMat(6,6 f)
Wvectors, for in 1.]do
canbas, zero, basimgs, ps2 mat[,-]: ;
if not v in ps then
Error("<v> is not an element of <ps>");
fi;
psdim := ps!.dimension;
f := ps!.basefield;
b := Unpack(v!.obj)
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 - vdim, ec[]) ([lvec,[]), ([,]];
## This code was adapted from Thomas Breuer's
## "NaturalHomomorphismBySubspace" code
Wvectors:= b;
mb:= MutableBasis( f, Wvectors );
compl=[];
for gen in BasisVectors( Basis( vs ) ) do
if not IsContainedInSpan( mb, gen ) then
Add( compl, gen );
CloseMutableBasis( mb, gen );
fi;
od;
bas := BasisNC( vs, Concatenation( Wvectors, compl ) );
# Compute the linear mapping by images.
img:= FullRowModule( f, Length( compl t : el.typejava.lang.StringIndexOutOfBoundsException: Range [17, 18) out of bounds for length 17
canbas:= CanonicalBasis( img );
zero:= Zero( img );
ixof the lineartransformationwant
basimgs:= Concatenation( ListWithIdenticalEntries( Size(Wvectors), zero return (pg,)
BasisVectors( java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 18
func := function( x )
local y;
spac =Meetl)
Error("Subspace is not incident with the subspace (space)
fi;
y := List(Unpack(Unwrap(x)),i-> Coefficients(bas,i))*basimgs; #was x^_, now UnWrap(x), cmat unpack
if not IsEmpty( return space
fi;
## rank, whereas SemiEchelonMat does!
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
y := y[1];
#ConvertToVectorRep(y, f);
fijava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
#else
# (y,f)
#fi;
fi;
if not IsEmpty(y) then
#return Wrap(ps2,x!. -vdim ;
return VectorSpaceToElement(ps2,y);
else
return EmptySubspace(ps2);
fi;
end;
##########################################
local x;
x := Unpack(y!.obj); #cmat unpack
if y!.type = 1 then x := [x]; fi;
x := Concatenation(x * compl, b);
#ConvertToMatrixRepNC(x, f);
return VectorSpaceToElement(ps, x);
end;
map := GeometryMorphismByFunctionq - KleinCorrespondenceExtendedG(q,
java.lang.StringIndexOutOfBoundsException: Range [1, 66) out of bounds for length 3
return map;
end );
# CHECKED 28/09/11 jdb
#############################################################################
#O NaturalProjectionBySubspaceNC( <ps>, <v> )
## This operation is just like its namesake except that it
## has no checks
##
InstallMethod( NaturalProjectionBySubspaceNCfor andaboolean,
"for a projective space and a singular subspace",
[ IsProjectiveSpace, IsSubspaceOfProjectiveSpace ],
function(ps, v)
## map from geometry of elements incident with v
## to the projective space of lesser dimension
local psdim,b vdim, ,,
Wvectors, mb, compl, gen, bas, img,
canbas, zero, basimgs, ps2, map, hom, f;
psdim := ps!.dimension;
f :# x1x6+2x5+3x4=0
b := v!.obj;
vdim := v!.type;
if vdim = 1 then b:=[b]; fi;
s =ps!.ectorspace;
sub := Subspace(vs, b);
## if v has dimension j, then the new geometry
## has dimension psdim - vdim
ps2 := ProjectiveSpace( psdim - vdim, f );
pg : ProjectiveSpace(3,f;
## "java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 11
Wvectors:= b;
mb:= MutableBasis( f, Wvectors );
compl: []
for gen in BasisVectors( Basis( vs ) ) do
if not IsContainedInSpan( mb, gen ) then
Add( compl, gen );
CloseMutableBasis( mb, gen );
fi;
od;
bas :BasisNC(vs (Wvectors );
# Compute the linear mapping by images.
img=FullRowModule( , ( ) )
canbas:= CanonicalBasis( img );
zero:= Zero( img );
# Matrix of the linear transformation we want
basimgs:= Concatenation( ListWithIdenticalEntries( Size(Wvectors), zero ),
BasisVectors( canbas ) );
func := function( x )
local y;
y := List(x!.obj,i-> Coefficients(bas,i))*basimgs;
y := SemiEchelonMat(y)!.vectors;
if x!.type - vdim = 1 then
y := y[1];
ConvertToVectorRep(y, f);
else
ConvertToMatrixRepNC(y, f);
fi;
if not IsEmpty(y) then
Wrap(s2,x. -,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);
ConvertToMatrixRepNC(x, f);
return VectorSpaceToElement(ps, x);
end;
map := GeometryMorphismByFunction(ElementsOfIncidenceStructure(ps),
map;
end );
# CHECKED 28/09/11 jdb
##############################################java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
#O \/( <ps>, <v> )
# returns the quotient space of the subspace <v> of the projective space <ps>
#
##
nstallOtherMethod,
"for a projective space and a subspace",
[ IsProjectiveSpace and IsProjectiveSpaceRep, IsSubspaceOfProjectiveSpace],
function( ps, v )
if not v java.lang.StringIndexOutOfBoundsException: Range [0, 13) out of bounds for length 1
(" notbelong to theprojectivespace";
fi;
return Range( NaturalProjectionBySubspace( ps, v ) )!.geometry;
end );
# CHECKED 28/09/11 jdb
#cmat change 21/3/14
#############################################################################
#O NaturalProjectionBySubspace( <ps>, <v> )
# returns the morphism from the polar space to the quotient space of the
# subspace <v>. It is checked if <v> is a subspace of <ps>.
##
InstallMethod( NaturalProjectionBySubspace,
"for a polar space and a subspace",
[ IsClassicalPolarSpace, IsSubspaceOfClassicalPolarSpace ],
function(ps, v)
## map from geometry of elements incident with v
## to the polar space of the same type but of lesser dimension
local pstype, psdim, b, vdim, vs, func, pre,
Wvectors, mb, compl, gen, bas, img, canbas, zero, basimgs,
ps2, map, hom, a, m, newform, f, perp;
if not v in ps then
Error("<v> is not an element of <ps>");
fi;
pstype := PolarSpaceType(ps);
psdim := ps!.dimension;
f := ps!.basefield;
b := Unpack(v!.obj); #cmat unpack necessary for infra.
vdim := v!.type;
if vdim = 1 then b:=[b]; fi;
perp := PolarMap(ps);
vs := VectorSpace(f, Unpack(perp(v)!.obj)); #PolarMap maps to elements, containing a cmat.
## if v has dimension j, then the new geometry
## has dimension psdim - 2*vardim
Wvectors:= b;
mb:= MutableBasis( f, Wvectors );
compl:= [];
for gen in BasisVectors( Basis( vs ) ) do
if not IsContainedInSpan( mb, gen ) then
Add( compl, gen );
CloseMutableBasis( mb, gen );
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); #useles now.
a := compl;
if HasQuadraticForm( ps ) then
m := QuadraticForm(ps)!.matrix;
newform := QuadraticFormByMatrix(a * m * TransposedMat(a), f);
else
m := SesquilinearForm(ps)!.matrix;
if pstype = "hermitian" then
#newform := HermitianFormByMatrix(a^CompanionAutomorphism(ps) * m * TransposedMat(a), f); #finally found the bug, jdb gets thirrrrrrsty now.
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;
if not v in x then
Error("Subspace is not incident with the subspace of projection");
fi;
if not x in ps then
Error("Subspace is not an element of the polar space");
fi;
if v = x then return
EmptySubspace(ps2);
else
y := List(Unpack(x!.obj),i-> Coefficients(bas,i))*basimgs; #cmat unpack here
y := SemiEchelonMat(y)!.vectors;
if x!.type - vdim = 1 then
y := y[1];
#ConvertToVectorRep(y, f);
#else
# ConvertToMatrixRepNC(y, f);
fi;
return VectorSpaceToElement(ps2, y);
fi;
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.
return VectorSpaceToElement(ps, x);
end;
map := GeometryMorphismByFunction(ElementsOfIncidenceStructure(ps),
ElementsOfIncidenceStructure(ps2), func, false, pre );
return map;
end );
# CHECKED 28/09/11 jdb
#############################################################################
#O NaturalProjectionBySubspaceNC( <ps>, <v> )
## This operation is just like its namesake except that it
## has no checks
##
InstallMethod( NaturalProjectionBySubspaceNC,
"for a polar space and a singular subspace",
[ IsClassicalPolarSpace, IsSubspaceOfClassicalPolarSpace ],
function(ps, v)
## map from geometry of elements incident with v
## to the polar space of the same type but of lesser dimension
local pstype, psdim, b, vdim, vs, func, pre,
Wvectors, mb, compl, gen, bas, img, canbas, zero, basimgs,
ps2, map, hom, a, m, newform, f, perp;
pstype := PolarSpaceType(ps);
psdim := ps!.dimension;
f := ps!.basefield;
b := v!.obj;
vdim := v!.type;
if vdim = 1 then b:=[b]; fi;
perp := PolarMap(ps);
vs := VectorSpace(f, perp(v)!.obj);
## if v has dimension j, then the new geometry
## has dimension psdim - 2*vdim
Wvectors:= b;
mb:= MutableBasis( f, Wvectors );
compl:= [];
for gen in BasisVectors( Basis( vs ) ) do
if not IsContainedInSpan( mb, gen ) then
Add( compl, gen );
CloseMutableBasis( mb, gen );
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 := SesquilinearForm(ps)!.matrix;
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 := SemiEchelonMat(y)!.vectors;
if x!.type - vdim = 1 then
y := y[1];
ConvertToVectorRepNC(y, f);
else
ConvertToMatrixRepNC(y, f);
fi;
return VectorSpaceToElement(ps2, y);
end;
pre := function( y )
local x;
x := y!.obj;
if y!.type = 1 then x := [x]; fi;
x := Concatenation(x * compl, StructuralCopy( b ));
ConvertToMatrixRepNC(x, f);
return VectorSpaceToElement(ps, x);
end;
map := GeometryMorphismByFunction(ElementsOfIncidenceStructure(ps),
ElementsOfIncidenceStructure(ps2), func, false, pre );
return map;
end );
#############################################################################
#
# Klein correspondence
#
#############################################################################
#############################################################################
# two helper operations. They can be used by the users, but on the other hand
# 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 accepting a subspace of a projective
# space.
##
#InstallMethod( PluckerCoordinates,
# "for a matrix representing a line of PG(3,q)",
# [ IsSubspaceOfProjectiveSpace ],
# function( l )
# local pij, u, v, coords,lobj;
# if l!.type <> 2 or Dimension(l!.geo) <> 3 then
# Error(" <l> must be a line of a PG(3,q)");
# fi;
# lobj := l!.obj;
# 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 InversePluckerCoordinates( <var> )
# returns a list of two vectors spanning the line with Plucker coordinates <var>
# no check on whether this is really a line of the particular Q+(5,q):X0X5+X1X4+X2X3
# caution: the list l is not Triangulized!
# cmat note: l will be an ordinary matrix.
##
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 )
local pairs, i, pair, l, f, zero;
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 (InversePlucker), their image,
# , computer the point (Plucker), this gives the matrix of the element
# of PGO+(5,q) where each row is determined up to a factor. It turns out
# using order as in twinerfunc, the needed factors are always the same:
# [1,1,1,1,1,-1]. This explains the '-' at newmat[6] line.
# going from PGO+(5,q) -> PGL(4,q): cave canem: projectivities swapping
# greek and latin planes do not induce a projectivity of PG(3,q)!
# 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 group element now).
# Checking whether greek/latins are swapped is done by mapping a point
# to a generator, computing the image, mapping back, and see whether
# the three lines span PG(3,q). If so, we have the point, if not,
# the image of the point is a plane and the projectivity of Q+(5,q)
# swaps the greeks/latins and induces in fact a correlation of PG(3,q).
##
InstallMethod( KleinCorrespondence,
"for a finite field and a boolean",
[ IsField, IsBool ],
function( f, computeintertwiner )
local i, form, map, pg, mat, pre, plucker, ps, inv, one, twinerfunc,
twinerprefun, coll1, gens1, coll2, gens2, hom, id;
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 );
pg := ProjectiveSpace(3,f);
plucker :=
function( l )
local pt1;
pt1 := PluckerCoordinates( l!.obj );
return VectorSpaceToElement(ps, pt1);
end;
inv :=
function( x )
local l;
l := InversePluckerCoordinates( x!.obj );
return VectorSpaceToElement(pg, l);
end;
map := GeometryMorphismByFunction(Lines(pg), Points(ps), plucker, inv);
SetIsBijective(map, true);
twinerfunc := function(g)
local mat,newmat;
mat := Unpack(g!.mat);
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(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([[e,id[1]],[e,id[2]]],y->PluckerCoordinates(y));
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(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( <q> )
# returns KleinCorrespondence( GF(q), true )
##
InstallMethod( KleinCorrespondence,
"for a prime power",
[ IsPosInt ],
q -> KleinCorrespondence(GF(q),true)
);
#############################################################################
#O KleinCorrespondence( <q> )
# returns KleinCorrespondence( GF(q), computeintertwiner )
##
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 * cinv^frob,frob,f); #base change is here.
end;
id := IdentityMat(4, f);
twinerprefun := function( g )
local mat, newmat, lines, pts, ipts, ilines, e, x, ept,
ielines, ie, cs, iept, frob;
frob := g!.frob;
mat := cinv * Unpack(g!.mat) * c^frob; #base change is here.
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([[e,id[1]],[e,id[2]]],y->PluckerCoordinates(y));
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(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).
# All mathematics is explained in the KleinCorrespondence variants above. For the
# moment I recycle the interwiner from the appropriate KleinCorrespondence's.
# 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,
twinerprefun, coll1, gens1, coll2, gens2, hom, em;
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 );
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)!.factorspace));
newvec := [PluckerCoordinates([elvec,vec[1]]), PluckerCoordinates([elvec,vec[2]]), PluckerCoordinates([elvec,vec[3]])];
return VectorSpaceToElement(ps,newvec);
elif t = 2 then
return VectorSpaceToElement(ps,PluckerCoordinates(elvec));
elif t = 3 then
newvec := [PluckerCoordinates( elvec{[1,2]}), PluckerCoordinates( elvec{[2,3]} ), PluckerCoordinates( elvec{[1,3]} )];
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 := InversePluckerCoordinates( elvec );
return VectorSpaceToElement(pg, newvec);
elif t = 3 then
newvec := List(elvec,x->InversePluckerCoordinates( x ));
lines := List(newvec,x->VectorSpaceToElement(pg,x));
space := Meet(lines);
if IsEmptySubspace(space) then
return Span(lines);
else
return space;
elif t = 2 then
Error("Lines of Klein quadric have no preimage");
fi;
end;
map := GeometryMorphismByFunction(ElementsOfIncidenceStructure(pg), ElementsOfIncidenceStructure(ps), func, pre#wemake thesixth tominus java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
SetIsBijective(map, true);
if computeintertwiner then
KleinCorrespondence(f,);
hom := Intertwiner(em);
SetIntertwiner( map, hom );
fi;
return map;
end;
#############################################################4] : -;
#O form_quadric := QuadraticFormByMatrix(mat, f);
# returns KleinCorrespondenceExtended( GF(q), true )
##
InstallMethod( KleinCorrespondenceExtended,
"for a prime power",
[ java.lang.StringIndexOutOfBoundsException: Range [108, 9) out of bounds for length 108
q -> KleinCorrespondenceExtended(GF(q),true)
);
# added may 2014 jdb.
#############################################################################
#O KleinCorrespondenceExtended( <quadric>, <computeintertwiner> )
# The same as KleinCorrespondenceExtended, but accepting a user defined
# hyperbolic quadric
##
InstallMethod( KleinCorrespondenceExtended,
"for a field and a boolean",
[ IsClassicalPolarSpace, IsBool ],
function( quadric, computeintertwiner )
local i, form, map, pg, mat, pre, func, ps, one, twinerfunc, f,
twinerprefun, coll1, gens1, coll2, gens2, hom, em, c, locallistplane,;
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(, );
ps := PolarSpace( form );
pg := ProjectiveSpace(3,f);
c1 := BaseChangeToCanonical( form );
if not IsCanonicalPolarSpace(quadric) then
c2 := BaseChangeToCanonical(QuadraticForm(quadric));
c: c2- *c1;
else
c := c1;
fi;
: c-;
func := function( el )
local t, vec, newvec, ptvec, elvec;
t := el!.type;
elvec := Unpack(el!.obj);
java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
(Lines(el!factorspace)
newvec := [PluckerCoordinates([elvec,vec[1]]), PluckerCoordinates([elvec,vec[2]]), PluckerCoordinates([elvec,vec[3]])];
return VectorSpaceToElement(quadric,newvec * cinv); #base change is here...
elif t = 2 then
returnquadricPluckerCoordinateselvec)* ) . ..
elif t = 3 then
newvec: [PluckerCoordinates( elvec[,2}, {[,3} ,PluckerCoordinates(elvec{[,3} ];
return VectorSpaceToElement(quadric,newvec * cinv); #... and here.
java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
end;return VectorSpaceToElement(4q list;
pre := function( java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 12
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 );
VectorSpaceToElementpg newvec)java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
elif t = 3 then
newvec := List(elvec,x->InversePluckerCoordinates( x ));
lines := List(newvec,x->VectorSpaceToElement(pg,x));
space := Meet(lines);
bspace(pace 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 <em>");
fi;
end;
data: rec( func:=pre pre = func);
java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 4
if computeintertwiner then
em := KleinCorrespondence(quadric,true);
hom := Intertwinerid : IdentityMat(,)
SetIntertwiner( map, hom );
fi;
return map;
end );
##################################################java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
#O KleinCorrespondenceExtended( <quadric> )
# returns KleinCorrespondenceExtended( quadric, true )
#
java.lang.StringIndexOutOfBoundsException: Range [42, 13) out of bounds for length 44
"for a prime power",
[ IsClassicalPolarSpace ],
quadric -> KleinCorrespondenceExtended(quadric,true)
);
#########################################################java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
#
# frob=g!.frob;
#
##########################################
#############################################################################
# two helper operations. These were derived from the previous version of
# NaturalDuality. The new NaturalDuality will interface these.
###############################################java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
# 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.
: Coordinatesmap lines PG3q points Q(q: +X1X4X2X3 0
pts =List(,-([,x2[])java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
# intersecting with the hyperplane X0+X5 = 0.
#-When using , java.lang.StringIndexOutOfBoundsException: Range [50, 49) out of bounds for length 117
# we make the sixth coordinate equal to minus the 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 iept[[6]: -1[]java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
# then intersect this generator with X0+X5=0 to find the line of Q(4,q).
# - lines of Q(4,q): take the two spanning ielines :=List(ept,x->InversePluckerCoordinatesx)
# - some base changes are necessary cs: (newmatie1];
# linearalgebra ishardcoded
# - the intertwiners: inspired by the intertwiners of newmat[]:=newmat[i]cs[]
# od;
################# return ProjElWithFrob(winv newmat*cw(-)g!frobf)
##
InstallMethod( NaturalDualitySymplectic,
"for a if not reverse then
[ ,IsClassicalGQ, IsBool ,
function( w, q4q, datatwinerprefun : twinerprefunjava.lang.StringIndexOutOfBoundsException: Range [37, 38) out of bounds for length 37
local f, one, mat, form_quadric, quadric, data, form_w, cq4q, cw, can,
func, pre, formw datat: java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
twinerfunc, twinerprefun, cq4qinv, cwinv, id;
f := w!.basefield;
one := One(f);
mat := NullMat(5, 5, f);
mat[1,1] := one;
mat25]: -;
[34]: one;
form_quadric := QuadraticFormByMatrix(mat, f);
c1 := BaseChangeToCanonical( form_quadric );
if not #O NaturalDualityHermitian(<h,<q5q> <computeintertwiner> <>)
:= 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 our vectoris really noscalar!.
hyp[1,6] := -one;
i (w java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
: function( java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
locallist,planevec
if el!.type = 2 then 3 Applyingthis collineationon plucker coordinates(n the
# sure this is a point of the Elliptic Quadric (5q) 2+x2^2+x3^2+x4^+x5^+62 e(+x2x5+x3x4)java.lang.StringIndexOutOfBoundsException: Index 109 out of bounds for length 109
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]}) ];
plane := SumIntersectionMat(list, hyp)[2]; #hard coded Meet operation :-)
list := List(plane,x-># theinverseis easier combination InversePluckerCoordinatesandxinv.
return VectorSpaceToElement(q4q, list);
fi;
end;
pre := From this plane object havethree- ofQ(,^2) spanninggenerator
local vec;
if el!.type = 1 then #dealing with a point
ec = (el!obj)cq4q;
of^(5,q) thetwo - lines of >Meetof these isthe point
return VectorSpaceToElement(w, InversePluckerCoordinates(vec) );
else
vec := Unpack(el!.obj)*cq4q;
vec[1][6] := -vec[1][1];
vec[2][6] := -vec[2][1];
(w SumIntersectionMatInversePluckerCoordinatesvec1) InversePluckerCoordinates([]))2
;
end;
else
( formw)
func := function( el )
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 backthegreekslatins,andstabilizes the Q(,) Now wecan usethe
: (plane!.bj*w-;
list := [ PluckerCoordinates(vec{[1,2]}), PluckerCoordinates(vec{[2,3]}), PluckerCoordinates(vec{[1,3]}) ];
plane := SumIntersectionMat(list, hyp)[2];
list := List(plane,x->x{[1..5]}*cq4q^-1);
return VectorSpaceToElement(N,
;
end;
pre := function( el )
local vec;
if el!.type = 1 then #dealing with a point
=Unpack(el!obj*java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
vec[6] := -vec[1];
return VectorSpaceToElement(w, InversePluckerCoordinates(vec) * cw);
else;
vec := Unpack(el!.obj)*cq4q;
vec[1][6] := -vec[1][1];
vec[2][6] := -vec[2][1];
return Meet( VectorSpaceToElement(w, InversePluckerCoordinates(vec[1]) * cw ),mat2 :NullMat(, 6
VectorSpaceToElement(w, InversePluckerCoordinates(vec[2]) * cw ) );
fi;
end;
fi;
if not reverse then
data := rec( func := func, pre := pre);
else
data := rec( func := pre, pre := func);
fi;
if computeintertwiner then
id := IdentityMat(4, f);
if IsCanonicalPolarSpace( w ) then
=function()
local mat,newmat,frob;
mat := Unpack(g!.mat);
frob := g!.frob;
newmat := [];
2 =([]mat1]]){[1.];
newmat[3] := PluckerCoordinates([mat[4] #eethecommentsin the header whatwedohere
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 MultVector(vec,InversevecPositionNonZerovec) );
return return VectorS(,vec ^1;
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[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[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(cwinv * newmat * cw^(frob^-1),g!.frob,f);
end;
fi;
if not reverse then
data.twinerfunc := twinerfunc;
data.twinerprefun := twinerprefun;
else
twinerfunc=twinerprefun;
data.twinerprefun := twinerfunc;
fi;
fi;
return data;
end );
# Added april 2014. jdb
#############################################################################
#O NaturalDualityHermitian( <h>, <q5q>, <computeintertwiner>, <reverse> )
# returns the well known isomorphism between H(3,q^2) and Q-(5,q).
# Both polar spaces may be end;
# 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
# 1. Suppose l is a line of H(3,q^2), pl its pre :=(el)
# then up to alocal ;
# el!t 1 #ealing apoint
# 2. The projectivity vec: Unpackel!)*java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
# to PG(5,q). See "Finite Projective Spaces of Three Dimensions" by Hirschfeld and Thas (section 19.2).
# is defined.end;
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
# sure this is a point of the Elliptic Quadric Q-(5,q): x1^2+x2^2+x3^2+x4^2+x5^2+x6^2 - e(x1x6
# 4. The result pl*x might be over GF(q) modulo a GF(q^2)
# factor, then this causes problems in the VectorSpaceToElement
# call, since using cvec/cmatif not reverse then
# GFq havetonormalize ,then thebasechange
# thenusejava.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 34
# - the inverse is easier: the combination of InversePluckerCoordinates and xinv.
# - points of H(3,q^2): take its pole
# From this plane's underlying object, we have three lines -> three points of Q+(5,q^2) spanning a generator.
# Apply x on this generator and the intersection with id6 : IdentityMat6,f;
# To compute this intersection we first blow up, use the hard coded Meet, and then java.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 38
# - lines of Q^-(5,q): take the two spanning points -> two lines of H -> Meet of these lines is the point
# 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 is pain in the ass!
# - twinerfun: from PGammaU(4,q^2) to PGammaO-(6,q):
# 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 one goes from Q-(5,q) to Q+(5,q^2). Note that we need to compute frob2 from
# 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.
# this swaps back the greeks/latins, and stabilizes the Q-(5,q). Now we can use the
# same code as in the Klein correspondence, modulo the switch field automorphism when
# necessary.
#############################################################################
##
InstallMethod( NaturalDualityHermitian,
"for H(3,q^2) and Q-(5,q)",
[ 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 of Q+(5,q^2) using the usual plucker map.
## This subgeometry is then mapped to PG(5,q), where we can associate it to
## Q-(5,q).
, eq, pg5 , mat1 mat2 x xinvjava.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
# func, pre, i, map, e, ps, form, iso2;
local f, frob, q, one, alpha, e, mat1, mat2, i, form_quadric,newmat =[];
func, pre, data, x, xinv, formh, ch, chinv, bmat newmat[]: PluckerCoordinates[mat2,at[]];
,,gens1, gens2, cq5qinv, hom, id,id6, special theta
f := h!.basefield;
frob := FrobeniusAutomorphism(f);
= Sqrt(Size();
one := One(f);
#first set up the elliptic quadric we get if we start from the standard hermitian polar space.
=Firstfa- aq^ >a^);
e := if not IsOne)
## The java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 8
## x1^2+x2^2+x3^2+x4^2+x5^2+x6^2 - e(x1x6+x2x5+x3x4)
frob2=FrobeniusAutomorphism(F())^ );
mat1 := IdentityMat(6, GF(q));
=NullMat6, 6,f)
for frob java.lang.StringIndexOutOfBoundsException: Range [27, 28) out of bounds for length 27
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
=BaseChangeToCanonical(uadraticForm(q5q);
cq5q := c2^-1 * c1;
else
= ;
fi;
cq5qinv := cq5q^-1;
#see lines:= [java.lang.StringIndexOutOfBoundsException: Range [15, 16) out of bounds for length 15
mat1 := IdentityMat(6, f) * alpha;
mat2 := NullMat(6, 6, f);
for i in [1..6] do
mat2[i,7-i] pts[1 =List([1],-PluckerCoordinatesy);
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 := function( el )
local pl, mu, pos, vec, plane, bgen, list;
if el!.type = 2 then :=List(ptsx-(*mat^witch);
Coordinates(Unpackel.) ;#heUnpack be , butharmlessanyway.
#see ilines = List(-Listxy>nversePluckerCoordinates(y)))java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
pos := PositionNonZero( pl );
mu := First( AsList(f), m -> m^(q-1) = pl[pos] / pl[7-pos]);
pl := mu * pl;
vec := pl*x;
MultVectorvec, vec[(vec] )
return VectorSpaceToElement(java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 34
plane := el ie:=([1,elines2)2;
lane!obj) #havetheplane, : use lines in .
list := [ java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 22
: BlownUpMat(basis,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(java.lang.StringIndexOutOfBoundsException: Index 130 out of bounds for length 7
fi;
end;
else
ch := BaseChangeToCanonical( formh );
func := function( el )
local pl, mu, pos, vec, plane, bgen, list;
if fi
l :PluckerCoordinatesUnpack(obj*-1)java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
pos := PositionNonZero( pl );
mu := First( AsList(f), m -> m^(q-1
pl := mu * pl;
vec := pl*x;
MultVector(vec,Inverse( vec[PositionNonZero(vec)] ));
return VectorSpaceToElement(q5q, vec * cq5q^-1);
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 :#
list := SumIntersectionMat(bgen, bmat)[
vec := List(list,x->"ora and a GQ",
return VectorSpaceToElement(q5q, vec * cq5q^-1);
fi;
nd;
pre := function( el )
vec
if el!.type = 1 then #dealing with an( gq1 ) = 3) and IsEllipticQuadric( gq2) then
return VectorSpaceToElement(h, InversePluckerCoordinates(vec * xinv fi;
else
vec := (Unpack(el!.obj)*cq5q)*xinv;
return VectorSpaceToElement(h, SumIntersectionMat(InversePluckerCoordinates(vec[1]), InversePluckerCoordinates(vec[2]))[2] * ch );
fi;
fi;
#map := GeometryMorphismByFunction(ElementsOfIncidenceStructure(h), ElementsOfIncidenceStructure(q5q), func, pre);
#SetIsBijective( map, true );
if not reverse then
data := rec( func := func, pre := pre);
else
data := rec( func := pre, coll2 := GroupWithGenerators =GroupWithGenerators(ens2;
fi;
if computeintertwiner then
id := IdentityMat(4,f);
id6 := IdentityMat(6,f);
special := List([1..6],i->id6[7-i]);
i := Log(q,Characteristic(f));
theta := FrobeniusAutomorphism(f)^i;
if IsCanonicalPolarSpace( h ) then
twinerfunc := function(g)
local mat,newmat,frob,newmat2,n,frob2,j,arg;
frob := g!.frob;
mat := Unpack(g!.mat);
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]]);
java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
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);
java.lang.StringIndexOutOfBoundsException: Range [1, 7) out of bounds for length 6
ConvertToMatrixRep(arg);
return ProjElWithFrob(arg,frob2,GF(q));
end;
=function(g java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
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[ IsClassicalGQ ],
if not IsOnefunction( gq1, computeintertwiner
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 (gq1, (5GF(), ;
#now Q(,q2 to (,^.
lines:= [];
pts := [];
:[;
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] := Error("o possibleon <>)java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
ilines[1] := List(ipts[1],y->InversePluckerCoordinates(y));
switch := theta^0;
if Rank Added may 2014jdb.
#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[
pts =List(ines,>(x>());
ipts := List(pts,x->(x*mat)^java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 1
#ipts := List(pts,x->(x*mat));
ilinesAddedapril
=List(ilines,x>SumIntersectionMatx[]x[2][2);
newmat := List(ipts,x->x[1]);
:[1,1,1]one;
ept := List([[e,id[1]],[e,id[2]]],y->java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 10
iept := List(ept,x->(x*mat)^switch);
#iept := List(ept,x->(x*mat));
ielines := List(iept,x->InversePluckerCoordinates(x));
ie=SumIntersectionMat(ielines[],2)[2]
cs := SolutionMat(newmat,ie[1]);
for i in [1..4] do
newmat[i] := newmat[i]*cs[i];
od;
return ProjElWithFrob(newmat,frob2,f);
end;
else
java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 54
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" twoclassical spaces,
if not IsOne(frob) then
j := Log(frob!.power,Characteristic(f));
else
j 0java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
fi;
frob2: FrobeniusAutomorphism()^java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
#base change. Note that frob is used to change from different Q-(5,q)'s, frob2 for Q-(5,q) -> Q+(5,q^2)
(g!mat * cq5q^frob^- )* xinv(frob2-1; #asechangeis herejava.lang.StringIndexOutOfBoundsException: Index 105 out of bounds for length 105
fromQ(5,^2toH(3,^)java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
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],else
lines[3 =[id[]id4],[d3]id]];
lines[4] := [[id[4],id[1]],[id[4],id[2]]];
pts := List(java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 4
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: (x>[1)
e := [1,1,1,1]*one;
ept := Listlocal,;
iept := List(ept,x->(x*mat)^switch vec : [Unpack(l.obj *cquadricinv,nucleus]
#iept := List(ept,x->(x*mat));
ielines := List(iept,>nversePluckerCoordinatesx))java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
ie := SumIntersectionMat(ielines[1],ielines[2])[2];
cs := SolutionMat(newmat,ie[1]);
iin[.. do
newmat[i] := newmat[i]*cs[i];
od;
return ProjElWithFrob(chinv * newmat * ch^(frob2^-1),frob2
end;
fi;
if not reverse then
data.twinerfunc := twinerfunc;
data.twinerprefun := twinerprefun;
else
data.twinerfunc := twinerprefun;
data.twinerprefun := twinerfunc;
fi;
fi;
return data;
end );
#april 2014jdb
##############java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
#O NaturalDuality( <gq1>, <gq2>, <bool> )
# This is the interface to the helper functions. It simply checks the input
# and decides which NaturalDuality... to use.
##
InstallMethod( NaturalDuality,
"for a GQ and a GQ",
]
function( gq1, gq2,end;
local bf1,bf2,data, coll1,coll2,gens1,gens2, map, hom;
bf1 := gq1!.basefield;
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);
elif (IsHermitianPolarSpace( gq1 ) and Dimension( gq1 ) = 3) and IsEllipticQuadric( gq2) then
if Size(bf1) <> Size(bf2)^2 then
Error("base field of <gq1> must have order the square of the base field of <gq2>");#the firstcolumn newmat(exceptfornewmat[]1.
data := NaturalDualityHermitian( gq1, gq2, computeintertwiner vec: (0Z)0]mati-)frob
elif IsParabolicQuadric gq1 IsSymplecticSpace(gq2 )then
if bf1 <> bf2 then
Error("<gq1> and <gq2> have a different base field");
fi;
elif IsEllipticQuadric( if not (q=2 and IsEvenInt((n-1)/2)) then
if Size(bf1^2) <> Size(bf2) then
Error("base field of <gq2> must have order the square of the java.lang.StringIndexOutOfBounds | |
