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#############################################################################
##
## group.gd 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
##
##
## Declaration stuff for groups
##
#############################################################################
DeclareGlobalFunction("MakeAllProjectivePoints");
DeclareGlobalFunction("IsFiningScalarMatrix");
## For later versions of GenSS, Max has changed the number of variables
## for the operation FindBasePointCandidates.
if PackageInfo("GenSS")[1]!.Version > "0.95" then
DeclareOperation( "FindBasePointCandidates", [ IsGroup, IsRecord, IsInt ] );
else
DeclareOperation( "FindBasePointCandidates", [ IsGroup, IsRecord, IsInt, IsObject ] );
fi;
###################################################################
# Construction operations for projective elements, that is matrices modulo scalars
# including representations
###################################################################
DeclareCategory( "IsProjGrpEl", IsComponentObjectRep and IsMultiplicativeElementWithInverse );
DeclareCategoryCollections( "IsProjGrpEl" );
InstallTrueMethod( IsGeneratorsOfMagmaWithInverses, IsProjGrpElCollection );
DeclareRepresentation( "IsProjGrpElRep", IsProjGrpEl, ["mat","fld"] );
BindGlobal( "ProjElsFamily",
NewFamily( "ProjElsFamily", IsObject, IsProjGrpEl ) );
BindGlobal( "ProjElsCollFamily", CollectionsFamily(ProjElsFamily) );
BindGlobal( "ProjElsType", NewType( ProjElsFamily,
IsProjGrpEl and IsProjGrpElRep) );
DeclareOperation( "ProjEl", [IsMatrix and IsFFECollColl] );
DeclareOperation( "ProjEls", [IsList] );
#InstallTrueMethod(IsHandledByNiceMonomorphism, IsProjectivityGroup);
DeclareOperation( "Projectivity", [ IsList, IsField] );
DeclareOperation( "Projectivity", [ IsProjectiveSpace, IsMatrix] );
###################################################################
# Construction operations for projective semilinear elements, that is matrices modulo scalars
# and a Frobenius automorphism, including representations
###################################################################
DeclareCategory( "IsProjGrpElWithFrob", IsComponentObjectRep and IsMultiplicativeElementWithInverse );
DeclareCategoryCollections( "IsProjGrpElWithFrob" );
InstallTrueMethod( IsGeneratorsOfMagmaWithInverses, IsProjGrpElWithFrobCollection );
DeclareRepresentation( "IsProjGrpElWithFrobRep", IsProjGrpElWithFrob, ["mat","fld","frob"] );
BindGlobal( "ProjElsWithFrobFamily",
NewFamily( "ProjElsWithFrobFamily",IsObject,IsProjGrpElWithFrob) );
BindGlobal( "ProjElsWithFrobCollFamily",
CollectionsFamily(ProjElsWithFrobFamily) );
BindGlobal( "ProjElsWithFrobType",
NewType( ProjElsWithFrobFamily,
IsProjGrpElWithFrob and IsProjGrpElWithFrobRep) );
DeclareOperation( "ProjElWithFrob", [IsMatrix and IsFFECollColl, IsMapping] );
DeclareOperation( "ProjElWithFrob", [IsMatrix and IsFFECollColl, IsMapping, IsField] );
DeclareOperation( "ProjElsWithFrob", [IsList] );
DeclareOperation( "ProjElsWithFrob", [IsList, IsField] );
#DeclareOperation( "ProjectiveSemilinearMap", [ IsList, IsField] ); # no longer valid (ml 8/11/12)
DeclareOperation( "CollineationOfProjectiveSpace", [ IsList, IsField] );
DeclareOperation( "CollineationOfProjectiveSpace", [ IsList, IsMapping, IsField] );
DeclareOperation( "CollineationOfProjectiveSpace", [ IsProjectiveSpace, IsMatrix] );
DeclareOperation( "CollineationOfProjectiveSpace", [ IsProjectiveSpace, IsMatrix, IsMapping] );
DeclareOperation( "CollineationOfProjectiveSpace", [ IsProjectiveSpace, IsMapping] );
DeclareOperation( "Collineation", [IsProjectiveSpace, IsMatrix] );
DeclareOperation( "Collineation", [IsProjectiveSpace, IsMatrix, IsMapping] );
DeclareOperation( "ProjectiveSemilinearMap", [ IsList, IsMapping, IsField] );
#DeclareSynonym( "CollineationOfProjectiveSpace", ProjectiveSemilinearMap); # no longer valid (ml 8/11/12)
DeclareOperation( "ProjectivityByImageOfStandardFrameNC", [IsProjectiveSpace, IsList] );
###################################################################
# Tests whether collineation is a projectivity and so on ...
###################################################################
DeclareProperty( "IsProjectivity", IsProjGrpEl );
DeclareProperty( "IsProjectivity", IsProjGrpElWithFrob );
DeclareProperty( "IsStrictlySemilinear", IsProjGrpEl );
DeclareProperty( "IsStrictlySemilinear", IsProjGrpElWithFrob );
DeclareProperty( "IsCollineation", IsProjGrpEl );
DeclareProperty( "IsCollineation", IsProjGrpElWithFrob );
###################################################################
# Some operations for elements
###################################################################
DeclareOperation( "MatrixOfCollineation", [ IsProjGrpElWithFrob and IsProjGrpElWithFrobRep ] );
DeclareOperation( "MatrixOfCollineation", [ IsProjGrpEl and IsProjGrpElRep] );
DeclareOperation( "FieldAutomorphism", [ IsProjGrpElWithFrob and IsProjGrpElWithFrobRep ] );
#################################################
# Frobenius automorphisms and groups using them:
#################################################
DeclareGlobalFunction( "OnProjPoints" );
DeclareGlobalFunction( "OnProjSubspacesNoFrob" );
# the following are not necessary, since the FinInG projectivity group is constructed
# as a projective semilinear group (i.e. a collineation group), and for these
# groups we have the operations defined (ml 05/11/2012)
#DeclareOperation( "ActionOnAllProjPoints", [IsProjectivityGroup] );
#DeclareAttribute( "Dimension", IsProjectivityGroup );
#DeclareProperty( "CanComputeActionOnPoints", IsProjectivityGroup );
#DeclareSynonym( "IsProjectiveSemilinearGroup", IsGroup and IsProjGrpElWithFrobCollection);
DeclareSynonym( "IsProjectiveGroupWithFrob", IsGroup and IsProjGrpElWithFrobCollection);
DeclareProperty( "IsProjectivityGroup", IsProjectiveGroupWithFrob );
DeclareProperty( "IsCollineationGroup", IsProjectiveGroupWithFrob );
#################################################
# action functions:
#################################################
InstallTrueMethod( IsHandledByNiceMonomorphism, IsProjectiveGroupWithFrob );
DeclareGlobalFunction( "OnProjSubspaces" );
DeclareGlobalFunction( "OnSetsProjSubspaces" );
DeclareGlobalFunction( "OnProjPointsWithFrob" );
DeclareGlobalFunction( "OnProjSubspacesWithFrob" );
DeclareOperation( "ActionOnAllProjPoints", [IsProjectiveGroupWithFrob] );
DeclareAttribute( "Dimension", IsProjectiveGroupWithFrob );
DeclareProperty( "CanComputeActionOnPoints", IsProjectiveGroupWithFrob );
DeclareGlobalFunction( "NiceMonomorphismByOrbit" );
DeclareGlobalFunction( "NiceMonomorphismByDomain" );
###########################
# Some group constructions:
###########################
## helper operations for canonical matrices, for classical groups.
DeclareOperation( "CanonicalGramMatrix", [IsString, IsPosInt, IsField]);
DeclareOperation( "CanonicalQuadraticForm", [IsString, IsPosInt, IsField]);
## The following are conjugates of the groups in the classical groups
## library which are compatible with the canonical forms in FinInG
DeclareOperation( "SOdesargues", [IsInt, IsPosInt, IsField and IsFinite]);
DeclareOperation( "GOdesargues", [IsInt, IsPosInt, IsField and IsFinite]);
DeclareOperation( "SUdesargues", [IsPosInt, IsField and IsFinite]);
DeclareOperation( "GUdesargues", [IsPosInt, IsField and IsFinite]);
DeclareOperation( "Spdesargues", [IsPosInt, IsField and IsFinite]);
## The following are methods which return the full group of invertible
## matrices which preserve a form up to scalars
DeclareOperation( "GeneralSymplecticGroup", [IsPosInt, IsField and IsFinite]);
DeclareOperation( "GSpdesargues", [IsPosInt, IsField and IsFinite]);
DeclareOperation( "DeltaOminus", [IsPosInt, IsField and IsFinite]);
DeclareOperation( "DeltaOplus", [IsPosInt, IsField and IsFinite]);
## The following are methods which return the full group of invertible
## matrices which preserve a form up to scalars and a field aut.
DeclareOperation( "GammaOminus", [IsPosInt, IsField and IsFinite]);
DeclareOperation( "GammaO", [IsPosInt, IsField and IsFinite]);
DeclareOperation( "GammaOplus", [IsPosInt, IsField and IsFinite]);
DeclareOperation( "GammaU", [IsPosInt, IsField and IsFinite]);
DeclareOperation( "GammaSp", [IsPosInt, IsField and IsFinite]);
[ Dauer der Verarbeitung: 0.34 Sekunden
(vorverarbeitet)
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