|
#############################################################################
##
## classic.gi 'Forms' package
##
## Provide the methods for 'GO', 'SO', 'Omega', 'GU', 'SU', and 'Sp' that
## involve a prescribed invariant form.
##
# Compatibility with GAP < 4.12
if not IsBound(IsMatrixOrMatrixObj) then
BindGlobal("IsMatrixOrMatrixObj", IsMatrixObj);
fi;
# We cannot use the function 'IsEqualProjective' from the recog package
# because the matrices that describe forms can have zero rows.
BindGlobal( "_IsEqualModScalars",
function( mat1, mat2 )
local m, n, i, j, s;
m:= NrRows( mat1 );
if m <> NrRows( mat2 ) then
return false;
fi;
n:= NrCols( mat1 );
if n <> NrCols( mat2 ) then
return false;
fi;
for i in [ 1 .. m ] do
for j in [ 1 .. n ] do
if not IsZero( mat1[ i, j ] ) then
s:= mat2[ i, j ] / mat1[ i, j ];
if IsZero( s ) then
return false;
elif IsRowListMatrix( mat1 ) and IsRowListMatrix( mat2 ) then
# separate case for performance reasons
return ForAll( [ 1 .. m ], i -> s * mat1[i] = mat2[i] );
fi;
return s * mat1 = mat2;
fi;
od;
od;
return IsZero( mat2 );
end );
BindGlobal("Forms_OrthogonalGroup",
function( g, form )
local stored, gf, d, wanted, mat1, mat2, mat, matinv, gens, gg;
stored:= InvariantQuadraticForm( g ).matrix;
# If the prescribed form fits then just return.
if stored = form!.matrix then
return g;
fi;
gf:= FieldOfMatrixGroup( g );
d:= DimensionOfMatrixGroup( g );
# Compute a base change matrix.
# (Check that the canonical forms are equal.)
wanted:= QuadraticFormByMatrix( stored, gf );
mat1:= BaseChangeToCanonical( form );
mat2:= BaseChangeToCanonical( wanted );
if not _IsEqualModScalars(
Forms_RESET( mat1 * form!.matrix * TransposedMat( mat1 ) ),
Forms_RESET( mat2 * stored * TransposedMat( mat2 ) ) ) then
Error( "canonical forms of <form> and <wanted> differ" );
fi;
mat:= mat2^-1 * mat1;
matinv:= mat^-1;
# Create the group w.r.t. the prescribed form.
gens:= List( GeneratorsOfGroup( g ), x -> matinv * x * mat );
gg:= GroupWithGenerators( gens );
UseIsomorphismRelation( g, gg );
if HasName( g ) then
SetName( gg, Name( g ) );
fi;
SetInvariantQuadraticForm( gg, rec( matrix:= form!.matrix ) );
if HasIsFullSubgroupGLorSLRespectingQuadraticForm( g ) then
SetIsFullSubgroupGLorSLRespectingQuadraticForm( gg,
IsFullSubgroupGLorSLRespectingQuadraticForm( g ) );
fi;
mat:= matinv * InvariantBilinearForm( g ).matrix * TransposedMat( matinv );
SetInvariantBilinearForm( gg, rec( matrix:= mat ) );
if Characteristic( gf ) <> 2 and
HasIsFullSubgroupGLorSLRespectingBilinearForm( g ) then
SetIsFullSubgroupGLorSLRespectingBilinearForm( gg,
IsFullSubgroupGLorSLRespectingBilinearForm( g ) );
fi;
return gg;
end );
#############################################################################
##
#O GeneralOrthogonalGroupCons( <filter>, <form> )
#O GeneralOrthogonalGroupCons( <filter>, <e>, <d>, <q>, <form> )
#O GeneralOrthogonalGroupCons( <filter>, <e>, <d>, <R>, <form> )
##
## 'GeneralOrthogonalGroup' is a plain function that is defined in the GAP
## library.
## It calls 'GeneralOrthogonalGroupCons',
## thus we have to declare the variants involving a quadratic form,
## and install the corresponding methods.
##
Perform(
[ IsMatrixOrMatrixObj, IsQuadraticForm, IsGroup and HasInvariantQuadraticForm ],
function( obj )
DeclareConstructor( "GeneralOrthogonalGroupCons",
[ IsGroup, obj ] );
DeclareConstructor( "GeneralOrthogonalGroupCons",
[ IsGroup, IsInt, IsPosInt, IsPosInt, obj ] );
DeclareConstructor( "GeneralOrthogonalGroupCons",
[ IsGroup, IsInt, IsPosInt, IsRing, obj ] );
end );
#############################################################################
##
#M GeneralOrthogonalGroupCons( <filt>, <form> )
##
InstallMethod( GeneralOrthogonalGroupCons,
"matrix group for matrix of form",
[ IsMatrixGroup and IsFinite, IsMatrixOrMatrixObj ],
{ filt, mat } -> GeneralOrthogonalGroupCons( filt,
QuadraticFormByMatrix( mat, BaseDomain( mat ) ) ) );
InstallMethod( GeneralOrthogonalGroupCons,
"matrix group for group with form",
[ IsMatrixGroup and IsFinite, IsGroup and HasInvariantQuadraticForm ],
{ filt, G } -> GeneralOrthogonalGroupCons( filt,
QuadraticFormByMatrix(
InvariantQuadraticForm( G ).matrix,
FieldOfMatrixGroup( G ) ) ) );
InstallMethod( GeneralOrthogonalGroupCons,
"matrix group for form",
[ IsMatrixGroup and IsFinite, IsQuadraticForm ],
function( filt, form )
local d, q, e;
d:= NumberRows( form!.matrix );
q:= Size( form!.basefield );
if IsOddInt( d ) then
e:= 0;
elif IsEllipticForm( form ) then
e:= -1;
else
e:= 1;
fi;
return GeneralOrthogonalGroupCons( filt, e, d, q, form );
end );
#############################################################################
##
#M GeneralOrthogonalGroupCons( <filt>, <e>, <d>, <q>, <form> )
##
InstallMethod( GeneralOrthogonalGroupCons,
"matrix group for <e>, dimension, finite field size, matrix of form",
[ IsMatrixGroup and IsFinite,
IsInt,
IsPosInt,
IsPosInt,
IsMatrixOrMatrixObj ],
{ filt, e, d, q, mat } -> GeneralOrthogonalGroupCons( filt, e, d, q,
QuadraticFormByMatrix( mat, GF(q) ) ) );
InstallMethod( GeneralOrthogonalGroupCons,
"matrix group for <e>, dimension, finite field size, group with form",
[ IsMatrixGroup and IsFinite,
IsInt,
IsPosInt,
IsPosInt,
IsGroup and HasInvariantQuadraticForm ],
{ filt, e, d, q, G } -> GeneralOrthogonalGroupCons( filt, e, d, q,
QuadraticFormByMatrix(
InvariantQuadraticForm( G ).matrix, GF(q) ) ) );
InstallMethod( GeneralOrthogonalGroupCons,
"matrix group for <e>, dimension, finite field size, form",
[ IsMatrixGroup and IsFinite,
IsInt,
IsPosInt,
IsPosInt,
IsQuadraticForm ],
function( filt, e, d, q, form )
local g, stored, wanted, mat1, mat2, mat, matinv, gens, gg;
# Create the default generators and form.
g:= GeneralOrthogonalGroupCons( filt, e, d, q );
return Forms_OrthogonalGroup( g, form );
end );
#############################################################################
##
#M GeneralOrthogonalGroupCons( <filt>, <e>, <d>, <R>, <form> )
##
InstallMethod( GeneralOrthogonalGroupCons,
"matrix group for <e>, dimension, finite field, matrix of form",
[ IsMatrixGroup and IsFinite,
IsInt,
IsPosInt,
IsField and IsFinite,
IsMatrixOrMatrixObj ],
{ filt, e, d, F, form } -> GeneralOrthogonalGroupCons( filt, e, d,
Size( F ),
QuadraticFormByMatrix( form, F ) ) );
InstallMethod( GeneralOrthogonalGroupCons,
"matrix group for <e>, dimension, finite field, group with form",
[ IsMatrixGroup and IsFinite,
IsInt,
IsPosInt,
IsField and IsFinite,
IsGroup and HasInvariantQuadraticForm ],
{ filt, e, d, F, G } -> GeneralOrthogonalGroupCons( filt, e, d,
Size( F ),
QuadraticFormByMatrix(
InvariantQuadraticForm( G ).matrix, F ) ) );
InstallMethod( GeneralOrthogonalGroupCons,
"matrix group for <e>, dimension, finite field, form",
[ IsMatrixGroup and IsFinite,
IsInt,
IsPosInt,
IsField and IsFinite,
IsQuadraticForm ],
{ filt, e, d, F, form } -> GeneralOrthogonalGroupCons( filt, e, d,
Size( F ), form ) );
#############################################################################
##
#O SpecialOrthogonalGroupCons( <filter>, <form> )
#O SpecialOrthogonalGroupCons( <filter>, <e>, <d>, <q>, <form> )
#O SpecialOrthogonalGroupCons( <filter>, <e>, <d>, <R>, <form> )
##
## 'SpecialOrthogonalGroup' is a plain function that is defined in the GAP
## library.
## It calls 'SpecialOrthogonalGroupCons',
## thus we have to declare the variants involving a quadratic form,
## and install the corresponding methods.
##
Perform(
[ IsMatrixOrMatrixObj, IsQuadraticForm, IsGroup and HasInvariantQuadraticForm ],
function( obj )
DeclareConstructor( "SpecialOrthogonalGroupCons",
[ IsGroup, obj ] );
DeclareConstructor( "SpecialOrthogonalGroupCons",
[ IsGroup, IsInt, IsPosInt, IsPosInt, obj ] );
DeclareConstructor( "SpecialOrthogonalGroupCons",
[ IsGroup, IsInt, IsPosInt, IsRing, obj ] );
end );
#############################################################################
##
#M SpecialOrthogonalGroupCons( <filt>, <form> )
##
InstallMethod( SpecialOrthogonalGroupCons,
"matrix group for matrix of form",
[ IsMatrixGroup and IsFinite, IsMatrixOrMatrixObj ],
{ filt, mat } -> SpecialOrthogonalGroupCons( filt,
QuadraticFormByMatrix( mat, BaseDomain( mat ) ) ) );
InstallMethod( SpecialOrthogonalGroupCons,
"matrix group for group with form",
[ IsMatrixGroup and IsFinite, IsGroup and HasInvariantQuadraticForm ],
{ filt, G } -> SpecialOrthogonalGroupCons( filt,
QuadraticFormByMatrix(
InvariantQuadraticForm( G ).matrix,
FieldOfMatrixGroup( G ) ) ) );
InstallMethod( SpecialOrthogonalGroupCons,
"matrix group for form",
[ IsMatrixGroup and IsFinite, IsQuadraticForm ],
function( filt, form )
local d, q, e;
d:= NumberRows( form!.matrix );
q:= Size( form!.basefield );
if IsOddInt( d ) then
e:= 0;
elif IsEllipticForm( form ) then
e:= -1;
else
e:= 1;
fi;
return SpecialOrthogonalGroupCons( filt, e, d, q, form );
end );
#############################################################################
##
#M SpecialOrthogonalGroupCons( <filt>, <e>, <d>, <q>, <form> )
##
InstallMethod( SpecialOrthogonalGroupCons,
"matrix group for <e>, dimension, finite field size, matrix of form",
[ IsMatrixGroup and IsFinite,
IsInt,
IsPosInt,
IsPosInt,
IsMatrixOrMatrixObj ],
{ filt, e, d, q, mat } -> SpecialOrthogonalGroupCons( filt, e, d, q,
QuadraticFormByMatrix( mat, GF(q) ) ) );
InstallMethod( SpecialOrthogonalGroupCons,
"matrix group for <e>, dimension, finite field size, group with form",
[ IsMatrixGroup and IsFinite,
IsInt,
IsPosInt,
IsPosInt,
IsGroup and HasInvariantQuadraticForm ],
{ filt, e, d, q, G } -> SpecialOrthogonalGroupCons( filt, e, d, q,
QuadraticFormByMatrix(
InvariantQuadraticForm( G ).matrix, GF(q) ) ) );
InstallMethod( SpecialOrthogonalGroupCons,
"matrix group for <e>, dimension, finite field size, form",
[ IsMatrixGroup and IsFinite,
IsInt,
IsPosInt,
IsPosInt,
IsQuadraticForm ],
function( filt, e, d, q, form )
local g, stored, wanted, mat1, mat2, mat, matinv, gens, gg;
# Create the default generators and form.
g:= SpecialOrthogonalGroupCons( filt, e, d, q );
return Forms_OrthogonalGroup( g, form );
end );
#############################################################################
##
#M SpecialOrthogonalGroupCons( <filt>, <e>, <d>, <R>, <form> )
##
InstallMethod( SpecialOrthogonalGroupCons,
"matrix group for <e>, dimension, finite field, matrix of form",
[ IsMatrixGroup and IsFinite,
IsInt,
IsPosInt,
IsField and IsFinite,
IsMatrixOrMatrixObj ],
{ filt, e, d, F, form } -> SpecialOrthogonalGroupCons( filt, e, d,
Size( F ),
QuadraticFormByMatrix( form, F ) ) );
InstallMethod( SpecialOrthogonalGroupCons,
"matrix group for <e>, dimension, finite field, group with form",
[ IsMatrixGroup and IsFinite,
IsInt,
IsPosInt,
IsField and IsFinite,
IsGroup and HasInvariantQuadraticForm ],
{ filt, e, d, F, G } -> SpecialOrthogonalGroupCons( filt, e, d,
Size( F ),
QuadraticFormByMatrix(
InvariantQuadraticForm( G ).matrix, F ) ) );
InstallMethod( SpecialOrthogonalGroupCons,
"matrix group for <e>, dimension, finite field, form",
[ IsMatrixGroup and IsFinite,
IsInt,
IsPosInt,
IsField and IsFinite,
IsQuadraticForm ],
{ filt, e, d, F, form } -> SpecialOrthogonalGroupCons( filt, e, d,
Size( F ), form ) );
#############################################################################
##
#O OmegaCons( <filt>, [<e>, <d>, <q>, ]<form> )
#O Omega( [<filt>, ]<form> )
#O Omega( [<filt>, ][<e>, ]<d>, <q>, <form> )
#O Omega( [<filt>, ][<e>, ]<d>, <R>, <form> )
##
## 'Omega' is an operation hat is defined in the GAP library.
## Thus we have to declare the variants involving a quadratic form,
## and install the corresponding 'Omega' methods that call 'OmegaCons'.
##
## Install the methods involving <form>, which may be either a matrix or
## a quadratic form or a group with stored 'InvariantQuadraticForm'.
## (The other methods have been installed in the GAP library.)
##
Perform(
[ IsMatrixOrMatrixObj, IsQuadraticForm, IsGroup and HasInvariantQuadraticForm ],
function( obj )
DeclareConstructor( "OmegaCons", [ IsGroup, obj ] );
DeclareConstructor( "OmegaCons", [ IsGroup, IsInt, IsPosInt, IsPosInt, obj ] );
DeclareOperation( "Omega", [ obj ] );
InstallMethod( Omega, [ obj ], x -> OmegaCons( IsMatrixGroup, x ) );
DeclareOperation( "Omega", [ IsFunction, obj ] );
InstallMethod( Omega, [ IsFunction, obj ], OmegaCons );
DeclareOperation( "Omega", [ IsPosInt, IsPosInt, obj ] );
InstallMethod( Omega, [ IsPosInt, IsPosInt, obj ],
{ d, q, x } -> OmegaCons( IsMatrixGroup, 0, d, q, x ) );
DeclareOperation( "Omega", [ IsPosInt, IsRing, obj ] );
InstallMethod( Omega, [ IsPosInt, IsField and IsFinite, obj ],
{ d, R, x } -> OmegaCons( IsMatrixGroup, 0, d, Size( R ), x ) );
DeclareOperation( "Omega", [ IsFunction, IsPosInt, IsPosInt, obj ] );
InstallMethod( Omega, [ IsFunction, IsPosInt, IsPosInt, obj ],
{ filt, d, q, x } -> OmegaCons( filt, 0, d, q, x ) );
DeclareOperation( "Omega", [ IsFunction, IsPosInt, IsRing, obj ] );
InstallMethod( Omega, [ IsFunction, IsPosInt, IsField and IsFinite, obj ],
{ filt, d, R, x } -> OmegaCons( filt, 0, d, Size( R ), x ) );
DeclareOperation( "Omega", [ IsInt, IsPosInt, IsPosInt, obj ] );
InstallMethod( Omega, [ IsInt, IsPosInt, IsPosInt, obj ],
{ e, d, q, x } -> OmegaCons( IsMatrixGroup, e, d, q, x ) );
DeclareOperation( "Omega", [ IsInt, IsPosInt, IsRing, obj ] );
InstallMethod( Omega, [ IsInt, IsPosInt, IsField and IsFinite, obj ],
{ e, d, R, x } -> OmegaCons( IsMatrixGroup, e, d, Size( R ), x ) );
DeclareOperation( "Omega", [ IsFunction, IsInt, IsPosInt, IsPosInt, obj ] );
InstallMethod( Omega, [ IsFunction, IsInt, IsPosInt, IsPosInt, obj ],
OmegaCons );
DeclareOperation( "Omega", [ IsFunction, IsInt, IsPosInt, IsRing, obj ] );
InstallMethod( Omega, [ IsFunction, IsInt, IsPosInt, IsField and IsFinite, obj ],
{ filt, e, d, R, x } -> OmegaCons( filt, e, d, Size( R ), x ) );
end );
#############################################################################
##
#M OmegaCons( <filt>, <form> )
##
InstallMethod( OmegaCons,
"matrix group for matrix of form",
[ IsMatrixGroup and IsFinite, IsMatrixOrMatrixObj ],
{ filt, mat } -> OmegaCons( filt,
QuadraticFormByMatrix( mat, BaseDomain( mat ) ) ) );
InstallMethod( OmegaCons,
"matrix group for group with form",
[ IsMatrixGroup and IsFinite, IsGroup and HasInvariantQuadraticForm ],
{ filt, G } -> OmegaCons( filt,
QuadraticFormByMatrix(
InvariantQuadraticForm( G ).matrix,
FieldOfMatrixGroup( G ) ) ) );
InstallMethod( OmegaCons,
"matrix group for form",
[ IsMatrixGroup and IsFinite, IsQuadraticForm ],
function( filt, form )
local d, q, e;
d:= NumberRows( form!.matrix );
q:= Size( form!.basefield );
if IsOddInt( d ) then
e:= 0;
elif IsEllipticForm( form ) then
e:= -1;
else
e:= 1;
fi;
return OmegaCons( filt, e, d, q, form );
end );
#############################################################################
##
#M OmegaCons( <filt>, <e>, <d>, <q>, <form> )
##
InstallMethod( OmegaCons,
"matrix group for <e>, dimension, finite field size, matrix of form",
[ IsMatrixGroup and IsFinite,
IsInt,
IsPosInt,
IsPosInt,
IsMatrixOrMatrixObj ],
{ filt, e, d, q, mat } -> OmegaCons( filt, e, d, q,
QuadraticFormByMatrix( mat, GF(q) ) ) );
InstallMethod( OmegaCons,
"matrix group for <e>, dimension, finite field size, group with form",
[ IsMatrixGroup and IsFinite,
IsInt,
IsPosInt,
IsPosInt,
IsGroup and HasInvariantQuadraticForm ],
{ filt, e, d, q, G } -> OmegaCons( filt, e, d, q,
QuadraticFormByMatrix(
InvariantQuadraticForm( G ).matrix, GF(q) ) ) );
InstallMethod( OmegaCons,
"matrix group for <e>, dimension, finite field size, form",
[ IsMatrixGroup and IsFinite,
IsInt,
IsPosInt,
IsPosInt,
IsQuadraticForm ],
function( filt, e, d, q, form )
local g, stored, wanted, mat1, mat2, mat, matinv, gens, gg;
# Create the default generators and form.
g:= OmegaCons( filt, e, d, q );
return Forms_OrthogonalGroup( g, form );
end );
#############################################################################
##
#O GeneralUnitaryGroupCons( <filter>, <form> )
#O GeneralUnitaryGroupCons( <filter>, <d>, <q>, <form> )
##
## 'GeneralUnitaryGroup' is a plain function that is defined in the GAP
## library.
## It calls 'GeneralUnitaryGroupCons',
## thus we have to declare the variants involving a quadratic form,
## and install the corresponding methods.
##
Perform(
[ IsMatrixOrMatrixObj, IsHermitianForm, IsGroup and HasInvariantSesquilinearForm ],
function( obj )
DeclareConstructor( "GeneralUnitaryGroupCons", [ IsGroup, obj ] );
DeclareConstructor( "GeneralUnitaryGroupCons",
[ IsGroup, IsPosInt, IsPosInt, obj ] );
end );
#############################################################################
##
#M GeneralUnitaryGroupCons( <filt>, <form> )
##
InstallMethod( GeneralUnitaryGroupCons,
"matrix group for matrix of form",
[ IsMatrixGroup and IsFinite, IsMatrixOrMatrixObj ],
function( filt, mat )
local q, form;
# Guess the intended field.
q:= Size( BaseDomain( mat ) );
if q = RootInt( q, 2 )^2 then
form:= HermitianFormByMatrix( mat, GF(q) );
else
form:= HermitianFormByMatrix( mat, GF(q^2) );
fi;
return GeneralUnitaryGroupCons( filt, form );
end );
InstallMethod( GeneralUnitaryGroupCons,
"matrix group for group with form",
[ IsMatrixGroup and IsFinite, IsGroup and HasInvariantSesquilinearForm ],
{ filt, G } -> GeneralUnitaryGroupCons( filt,
HermitianFormByMatrix(
InvariantSesquilinearForm( G ).matrix,
FieldOfMatrixGroup( G ) ) ) );
InstallMethod( GeneralUnitaryGroupCons,
"matrix group for form",
[ IsMatrixGroup and IsFinite, IsHermitianForm ],
{ filt, form } -> GeneralUnitaryGroupCons( filt,
NumberRows( form!.matrix ),
RootInt( Size( form!.basefield ), 2 ), form ) );
#############################################################################
##
#F TransposedFrobeniusMat( <mat>, <qq> )
## Helper function
##
InstallGlobalFunction( TransposedFrobeniusMat,
function( mat, qq )
local i, j;
mat:=MutableTransposedMat(mat);
for i in [ 1 .. NrRows(mat) ] do
for j in [ 1 .. NrCols(mat) ] do
mat[i,j] := mat[i,j]^qq;
od;
od;
return mat;
end );
#############################################################################
##
#M GeneralUnitaryGroupCons( <filt>, <d>, <q>, <form> )
##
InstallMethod( GeneralUnitaryGroupCons,
"matrix group for dimension, finite field size, matrix of form",
[ IsMatrixGroup and IsFinite,
IsPosInt,
IsPosInt,
IsMatrixOrMatrixObj ],
{ filt, d, q, mat } -> GeneralUnitaryGroupCons( filt, d, q,
HermitianFormByMatrix( mat, GF(q^2) ) ) );
InstallMethod( GeneralUnitaryGroupCons,
"matrix group for dimension, finite field size, group with form",
[ IsMatrixGroup and IsFinite,
IsPosInt,
IsPosInt,
IsGroup and HasInvariantSesquilinearForm ],
{ filt, d, q, G } -> GeneralUnitaryGroupCons( filt, d, q,
HermitianFormByMatrix(
InvariantSesquilinearForm( G ).matrix, GF(q^2) ) ) );
InstallMethod( GeneralUnitaryGroupCons,
"matrix group for dimension, finite field size, form",
[ IsMatrixGroup and IsFinite,
IsPosInt,
IsPosInt,
IsHermitianForm ],
function( filt, d, q, form )
local g, stored, wanted, mat1, mat2, mat, matinv, gens, gg;
# Create the default generators and form.
g:= GeneralUnitaryGroupCons( filt, d, q );
stored:= InvariantSesquilinearForm( g ).matrix;
# If the prescribed form fits then just return.
if stored = form!.matrix then
return g;
fi;
# Compute a base change matrix.
# (Check that the canonical forms are equal.)
wanted:= HermitianFormByMatrix( stored, GF(q^2) );
mat1:= BaseChangeToCanonical( form );
mat2:= BaseChangeToCanonical( wanted );
if mat1 * form!.matrix * TransposedFrobeniusMat( mat1, q ) <>
mat2 * stored * TransposedFrobeniusMat( mat2, q ) then
Error( "canonical forms of <form> and <wanted> differ" );
fi;
mat:= mat2^-1 * mat1;
matinv:= mat^-1;
# Create the group w.r.t. the prescribed form.
gens:= List( GeneratorsOfGroup( g ), x -> matinv * x * mat );
gg:= GroupWithGenerators( gens );
UseIsomorphismRelation( g, gg );
if HasName( g ) then
SetName( gg, Name( g ) );
fi;
SetInvariantSesquilinearForm( gg, rec( matrix:= form!.matrix ) );
if HasIsFullSubgroupGLorSLRespectingSesquilinearForm( g ) then
SetIsFullSubgroupGLorSLRespectingSesquilinearForm( gg,
IsFullSubgroupGLorSLRespectingSesquilinearForm( g ) );
fi;
return gg;
end );
#############################################################################
##
#O SpecialUnitaryGroupCons( <filter>, <form> )
#O SpecialUnitaryGroupCons( <filter>, <d>, <q>, <form> )
##
## 'SpecialUnitaryGroup' is a plain function that is defined in the GAP
## library.
## It calls 'SpecialUnitaryGroupCons',
## thus we have to declare the variants involving a quadratic form,
## and install the corresponding methods.
##
Perform(
[ IsMatrixOrMatrixObj, IsHermitianForm, IsGroup and HasInvariantSesquilinearForm ],
function( obj )
DeclareConstructor( "SpecialUnitaryGroupCons", [ IsGroup, obj ] );
DeclareConstructor( "SpecialUnitaryGroupCons",
[ IsGroup, IsPosInt, IsPosInt, obj ] );
end );
#############################################################################
##
#M SpecialUnitaryGroupCons( <filt>, <form> )
##
InstallMethod( SpecialUnitaryGroupCons,
"matrix group for matrix of form",
[ IsMatrixGroup and IsFinite, IsMatrixOrMatrixObj ],
function( filt, mat )
local q, form;
# Guess the intended field.
q:= Size( BaseDomain( mat ) );
if q = RootInt( q, 2 )^2 then
form:= HermitianFormByMatrix( mat, GF(q) );
else
form:= HermitianFormByMatrix( mat, GF(q^2) );
fi;
return SpecialUnitaryGroupCons( filt, form );
end );
InstallMethod( SpecialUnitaryGroupCons,
"matrix group for group with form",
[ IsMatrixGroup and IsFinite, IsGroup and HasInvariantSesquilinearForm ],
{ filt, G } -> SpecialUnitaryGroupCons( filt,
HermitianFormByMatrix(
InvariantSesquilinearForm( G ).matrix,
FieldOfMatrixGroup( G ) ) ) );
InstallMethod( SpecialUnitaryGroupCons,
"matrix group for form",
[ IsMatrixGroup and IsFinite, IsHermitianForm ],
{ filt, form } -> SpecialUnitaryGroupCons( filt,
NumberRows( form!.matrix ),
RootInt( Size( form!.basefield ), 2 ), form ) );
#############################################################################
##
#M SpecialUnitaryGroupCons( <filt>, <d>, <q>, <form> )
##
InstallMethod( SpecialUnitaryGroupCons,
"matrix group for dimension, finite field size, matrix of form",
[ IsMatrixGroup and IsFinite,
IsPosInt,
IsPosInt,
IsMatrixOrMatrixObj ],
{ filt, d, q, mat } -> SpecialUnitaryGroupCons( filt, d, q,
HermitianFormByMatrix( mat, GF(q^2) ) ) );
InstallMethod( SpecialUnitaryGroupCons,
"matrix group for dimension, finite field size, group with form",
[ IsMatrixGroup and IsFinite,
IsPosInt,
IsPosInt,
IsGroup and HasInvariantSesquilinearForm ],
{ filt, d, q, G } -> SpecialUnitaryGroupCons( filt, d, q,
HermitianFormByMatrix(
InvariantSesquilinearForm( G ).matrix, GF(q^2) ) ) );
InstallMethod( SpecialUnitaryGroupCons,
"matrix group for dimension, finite field size, form",
[ IsMatrixGroup and IsFinite,
IsPosInt,
IsPosInt,
IsHermitianForm ],
function( filt, d, q, form )
local g, stored, wanted, mat1, mat2, mat, matinv, gens, gg;
# Create the default generators and form.
g:= SpecialUnitaryGroupCons( filt, d, q );
stored:= InvariantSesquilinearForm( g ).matrix;
# If the prescribed form fits then just return.
if stored = form!.matrix then
return g;
fi;
# Compute a base change matrix.
# (Check that the canonical forms are equal.)
wanted:= HermitianFormByMatrix( stored, GF(q^2) );
mat1:= BaseChangeToCanonical( form );
mat2:= BaseChangeToCanonical( wanted );
if mat1 * form!.matrix * TransposedFrobeniusMat( mat1, q ) <>
mat2 * stored * TransposedFrobeniusMat( mat2, q ) then
Error( "canonical forms of <form> and <wanted> differ" );
fi;
mat:= mat2^-1 * mat1;
matinv:= mat^-1;
# Create the group w.r.t. the prescribed form.
gens:= List( GeneratorsOfGroup( g ), x -> matinv * x * mat );
gg:= GroupWithGenerators( gens );
UseIsomorphismRelation( g, gg );
if HasName( g ) then
SetName( gg, Name( g ) );
fi;
SetInvariantSesquilinearForm( gg, rec( matrix:= form!.matrix ) );
if HasIsFullSubgroupGLorSLRespectingSesquilinearForm( g ) then
SetIsFullSubgroupGLorSLRespectingSesquilinearForm( gg,
IsFullSubgroupGLorSLRespectingSesquilinearForm( g ) );
fi;
return gg;
end );
#############################################################################
##
#O SymplecticGroupCons( <filter>, <form> )
#O SymplecticGroupCons( <filter>, <d>, <q>, <form> )
#O SymplecticGroupCons( <filter>, <d>, <R>, <form> )
##
## 'SymplecticGroup' is a plain function that is defined in the GAP
## library.
## It calls 'SymplecticGroupCons',
## thus we have to declare the variants involving a quadratic form,
## and install the corresponding methods.
##
Perform(
[ IsMatrixOrMatrixObj, IsBilinearForm, IsGroup and HasInvariantBilinearForm ],
function( obj )
DeclareConstructor( "SymplecticGroupCons", [ IsGroup, obj ] );
DeclareConstructor( "SymplecticGroupCons",
[ IsGroup, IsPosInt, IsPosInt, obj ] );
DeclareConstructor( "SymplecticGroupCons",
[ IsGroup, IsPosInt, IsRing, obj ] );
end );
#############################################################################
##
#M SymplecticGroupCons( <filt>, <form> )
##
InstallMethod( SymplecticGroupCons,
"matrix group for matrix of form",
[ IsMatrixGroup and IsFinite, IsMatrixOrMatrixObj ],
{ filt, mat } -> SymplecticGroupCons( filt,
BilinearFormByMatrix( mat, BaseDomain( mat ) ) ) );
InstallMethod( SymplecticGroupCons,
"matrix group for group with form",
[ IsMatrixGroup and IsFinite, IsGroup and HasInvariantBilinearForm ],
{ filt, G } -> SymplecticGroupCons( filt,
BilinearFormByMatrix(
InvariantBilinearForm( G ).matrix,
FieldOfMatrixGroup( G ) ) ) );
InstallMethod( SymplecticGroupCons,
"matrix group for form",
[ IsMatrixGroup and IsFinite, IsBilinearForm ],
{ filt, form } -> SymplecticGroupCons( filt,
NumberRows( form!.matrix ),
Size( form!.basefield ), form ) );
#############################################################################
##
#M SymplecticGroupCons( <filt>, <d>, <q>, <form> )
##
InstallMethod( SymplecticGroupCons,
"matrix group for dimension, finite field size, matrix of form",
[ IsMatrixGroup and IsFinite,
IsPosInt,
IsPosInt,
IsMatrixOrMatrixObj ],
{ filt, d, q, mat } -> SymplecticGroupCons( filt, d, q,
BilinearFormByMatrix( mat, GF(q) ) ) );
InstallMethod( SymplecticGroupCons,
"matrix group for dimension, finite field size, group with form",
[ IsMatrixGroup and IsFinite,
IsPosInt,
IsPosInt,
IsGroup and HasInvariantBilinearForm ],
{ filt, d, q, G } -> SymplecticGroupCons( filt, d, q,
BilinearFormByMatrix(
InvariantBilinearForm( G ).matrix, GF(q) ) ) );
InstallMethod( SymplecticGroupCons,
"matrix group for dimension, finite field size, form",
[ IsMatrixGroup and IsFinite,
IsPosInt,
IsPosInt,
IsBilinearForm ],
function( filt, d, q, form )
local g, stored, wanted, mat1, mat2, mat, matinv, gens, gg;
# Create the default generators and form.
g:= SymplecticGroupCons( filt, d, q );
stored:= InvariantBilinearForm( g ).matrix;
# If the prescribed form fits then just return.
if stored = form!.matrix then
return g;
fi;
# Compute a base change matrix.
# (Check that the canonical forms are equal.)
wanted:= BilinearFormByMatrix( stored, GF(q) );
mat1:= BaseChangeToCanonical( form );
mat2:= BaseChangeToCanonical( wanted );
if mat1 * form!.matrix * TransposedMat( mat1 ) <>
mat2 * stored * TransposedMat( mat2 ) then
Error( "canonical forms of <form> and <wanted> differ" );
fi;
mat:= mat2^-1 * mat1;
matinv:= mat^-1;
# Create the group w.r.t. the prescribed form.
gens:= List( GeneratorsOfGroup( g ), x -> matinv * x * mat );
gg:= GroupWithGenerators( gens );
UseIsomorphismRelation( g, gg );
if HasName( g ) then
SetName( gg, Name( g ) );
fi;
SetInvariantBilinearForm( gg, rec( matrix:= form!.matrix ) );
if HasIsFullSubgroupGLorSLRespectingBilinearForm( g ) then
SetIsFullSubgroupGLorSLRespectingBilinearForm( gg,
IsFullSubgroupGLorSLRespectingBilinearForm( g ) );
fi;
return gg;
end );
#############################################################################
##
#M SymplecticGroupCons( <filt>, <d>, <R>, <form> )
##
InstallMethod( SymplecticGroupCons,
"matrix group for dimension, finite field, matrix of form",
[ IsMatrixGroup and IsFinite,
IsPosInt,
IsField and IsFinite,
IsMatrixOrMatrixObj ],
{ filt, d, F, form } -> SymplecticGroupCons( filt, d, Size( F ),
BilinearFormByMatrix( form, F ) ) );
InstallMethod( SymplecticGroupCons,
"matrix group for dimension, finite field, group with form",
[ IsMatrixGroup and IsFinite,
IsPosInt,
IsField and IsFinite,
IsGroup and HasInvariantBilinearForm ],
{ filt, d, F, G } -> SymplecticGroupCons( filt, d, Size( F ),
BilinearFormByMatrix(
InvariantBilinearForm( G ).matrix, F ) ) );
InstallMethod( SymplecticGroupCons,
"matrix group for dimension, finite field, form",
[ IsMatrixGroup and IsFinite,
IsPosInt,
IsField and IsFinite,
IsBilinearForm ],
{ filt, d, F, form } -> SymplecticGroupCons( filt, d, Size( F ), form ) );
[ Dauer der Verarbeitung: 0.39 Sekunden
(vorverarbeitet)
]
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