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#############################################################################
##
## correlations.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 correlation groups and polarities of a projective
## space.
##
#############################################################################
DeclareCategory( "IsProjectiveSpaceIsomorphism", IsSPGeneralMapping );
DeclareCategory( "IsStandardDualityOfProjectiveSpace", IsProjectiveSpaceIsomorphism );
DeclareCategory( "IsIdentityMappingOfElementsOfProjectiveSpace", IsProjectiveSpaceIsomorphism );
DeclareCategory( "IsProjGrpElWithFrobWithPSIsom", IsComponentObjectRep and IsMultiplicativeElementWithInverse );
DeclareSynonym( "IsCorrelationCollineation", IsProjGrpElWithFrobWithPSIsom);
DeclareProperty( "IsCorrelation", IsProjGrpElWithFrobWithPSIsom );# user-friendly functionality
DeclareProperty( "IsCorrelation", IsProjGrpElWithFrob );# user-friendly functionality
DeclareProperty( "IsCorrelation", IsProjGrpEl );# user-friendly functionality
#DeclareProperty( "IsCollineation", IsProjGrpElWithFrobWithPSIsom );
DeclareCategoryCollections( "IsProjGrpElWithFrobWithPSIsom" );
InstallTrueMethod( IsGeneratorsOfMagmaWithInverses, IsProjGrpElWithFrobWithPSIsomCollection );
DeclareRepresentation( "IsProjGrpElWithFrobWithPSIsomRep", IsProjGrpElWithFrobWithPSIsom, ["mat","fld","frob","psisom"] );
BindGlobal( "ProjElsWithFrobWithPSIsomFamily",
NewFamily( "ProjElsWithFrobWithPSIsomFamily",IsObject,
IsProjGrpElWithFrobWithPSIsom ) );
BindGlobal( "ProjElsWithFrobWithPSIsomCollFamily",
CollectionsFamily(ProjElsWithFrobWithPSIsomFamily) );
BindGlobal( "ProjElsWithFrobWithPSIsomType",
NewType( ProjElsWithFrobWithPSIsomFamily,
IsProjGrpElWithFrobWithPSIsom and
IsProjGrpElWithFrobWithPSIsomRep ) );
InstallTrueMethod( IsHandledByNiceMonomorphism, IsProjectiveGroupWithFrob );
DeclareSynonym( "IsProjGroupWithFrobWithPSIsom", IsGroup and IsProjGrpElWithFrobWithPSIsomCollection);
DeclareSynonym( "IsCorrelationCollineationGroup", IsGroup and IsProjGrpElWithFrobWithPSIsomCollection);
InstallTrueMethod( IsHandledByNiceMonomorphism, IsProjGroupWithFrobWithPSIsom );
DeclareGlobalFunction( "OnProjPointsWithFrobWithPSIsom" );
DeclareGlobalFunction( "OnProjSubspacesWithFrobWithPSIsom" );
DeclareGlobalFunction( "OnProjSubspacesExtended" );
DeclareOperation( "StandardDualityOfProjectiveSpace", [IsProjectiveSpace] );
DeclareOperation( "IdentityMappingOfElementsOfProjectiveSpace", [IsProjectiveSpace] );
DeclareOperation( "ActionOnAllPointsHyperplanes", [IsProjGroupWithFrobWithPSIsom] );
DeclareOperation( "ProjElWithFrobWithPSIsom",
[IsMatrix and IsFFECollColl, IsMapping, IsField] );
DeclareOperation( "ProjElWithFrobWithPSIsom",
[IsMatrix and IsFFECollColl, IsMapping, IsField,
IsStandardDualityOfProjectiveSpace] );
DeclareOperation( "ProjElWithFrobWithPSIsom",
[IsMatrix and IsFFECollColl, IsMapping, IsField,
IsGeneralMapping and IsSPGeneralMapping and IsOne] );
DeclareOperation( "ProjElsWithFrobWithPSIsom", [IsList, IsField] );
DeclareAttribute( "Dimension", IsProjGroupWithFrobWithPSIsom );
DeclareProperty( "CanComputeActionOnPoints", IsProjGroupWithFrobWithPSIsom );
DeclareOperation( "CorrelationOfProjectiveSpace", [ IsList, IsField] );
DeclareOperation( "CorrelationOfProjectiveSpace", [ IsList, IsMapping, IsField] );
DeclareOperation( "CorrelationOfProjectiveSpace", [ IsList, IsField, IsStandardDualityOfProjectiveSpace] );
DeclareOperation( "CorrelationOfProjectiveSpace", [ IsList, IsField, IsIdentityMappingOfElementsOfProjectiveSpace] );
DeclareOperation( "CorrelationOfProjectiveSpace", [ IsList, IsMapping, IsField, IsStandardDualityOfProjectiveSpace] );
DeclareOperation( "CorrelationOfProjectiveSpace", [ IsList, IsMapping, IsField, IsIdentityMappingOfElementsOfProjectiveSpace] );
DeclareOperation( "CorrelationOfProjectiveSpace", [ IsProjectiveSpace, IsMatrix, IsMapping, IsStandardDualityOfProjectiveSpace] );
DeclareOperation( "CorrelationOfProjectiveSpace", [ IsProjectiveSpace, IsMatrix, IsMapping, IsIdentityMappingOfElementsOfProjectiveSpace] );
DeclareOperation( "Correlation", [ IsProjectiveSpace, IsMatrix, IsMapping, IsStandardDualityOfProjectiveSpace] );
DeclareOperation( "Correlation", [ IsProjectiveSpace, IsMatrix, IsMapping, IsIdentityMappingOfElementsOfProjectiveSpace] );
DeclareOperation( "MatrixOfCorrelation", [ IsProjGrpElWithFrobWithPSIsom and IsProjGrpElWithFrobWithPSIsomRep ] );
DeclareOperation( "FieldAutomorphism", [ IsProjGrpElWithFrobWithPSIsom and IsProjGrpElWithFrobWithPSIsomRep ] );
DeclareOperation( "ProjectiveSpaceIsomorphism", [ IsProjGrpElWithFrobWithPSIsom and IsProjGrpElWithFrobWithPSIsomRep ] );
###################################################################
# Tests whether IsProjGrpElWithFrobWithPSIsom is a projectivity and so on ...
###################################################################
DeclareProperty( "IsProjectivity", IsProjGrpElWithFrobWithPSIsom );
DeclareProperty( "IsStrictlySemilinear", IsProjGrpElWithFrobWithPSIsom );
DeclareProperty( "IsCollineation", IsProjGrpElWithFrobWithPSIsom );
###################################################################
# Tests whether CorrelationCollineationGroup is a ProjectivityGroup and so on ...
###################################################################
DeclareProperty( "IsProjectivityGroup", IsProjGroupWithFrobWithPSIsom );
DeclareProperty( "IsCollineationGroup", IsProjGroupWithFrobWithPSIsom );
#############################################################################
#
# operations to create incidence preserving and reversing maps (to be placed in geometry.gd afterwards).
#
#############################################################################
#DeclareOperation( "Collineation", [IsIncidenceStructure, IsMultiplicativeElementWithInverse] );
#DeclareOperation( "Correlation", [IsIncidenceStructure, IsMultiplicativeElementWithInverse] );
#DeclareOperation( "Triality", [IsIncidenceStructure, IsMultiplicativeElementWithInverse] );
#DeclareOperation( "Polarity", [IsIncidenceStructure, IsMultiplicativeElementWithInverse] );
#DeclareOperation( "PolarityOfProjectiveSpace", [IsProjectiveSpace, IsMultiplicativeElementWithInverse] );
DeclareCategory( "IsPolarityOfProjectiveSpace", IsProjGrpElWithFrobWithPSIsomRep );
DeclareRepresentation( "IsPolarityOfProjectiveSpaceRep", IsProjGrpElWithFrobWithPSIsomRep, ["mat","fld","frob","psisom", "form"] );
#############################################################################
# polarities are equivalent with sesquilinear forms.
# This explains the basic constructor
#############################################################################
DeclareOperation( "PolarityOfProjectiveSpaceOp", [IsForm] );
DeclareOperation( "PolarityOfProjectiveSpace", [IsForm] );
DeclareOperation( "PolarityOfProjectiveSpace", [IsMatrix,IsField and IsFinite] );
DeclareOperation( "PolarityOfProjectiveSpace", [IsMatrix,IsFrobeniusAutomorphism,IsField and IsFinite] );
DeclareOperation( "HermitianPolarityOfProjectiveSpace", [IsMatrix, IsField and IsFinite ] );
DeclareOperation( "PolarityOfProjectiveSpace", [IsClassicalPolarSpace] );
#DeclareAttribute( "IsDegeneratePolarity", IsPolarityOfProjectiveSpace ); #obsolete.
DeclareOperation( "BaseField", [ IsPolarityOfProjectiveSpace ]);
DeclareAttribute( "GramMatrix", IsPolarityOfProjectiveSpace );
DeclareAttribute( "CompanionAutomorphism", IsPolarityOfProjectiveSpace );
DeclareAttribute( "SesquilinearForm", IsPolarityOfProjectiveSpace );
#############################################################################
# operations and attributes for polarities of projective space.
#############################################################################
DeclareProperty( "IsHermitianPolarityOfProjectiveSpace", IsPolarityOfProjectiveSpace );
DeclareProperty( "IsSymplecticPolarityOfProjectiveSpace", IsPolarityOfProjectiveSpace );
DeclareProperty( "IsOrthogonalPolarityOfProjectiveSpace", IsPolarityOfProjectiveSpace );
DeclareProperty( "IsPseudoPolarityOfProjectiveSpace", IsPolarityOfProjectiveSpace );
DeclareOperation( "IsAbsoluteElement", [ IsElementOfIncidenceStructure, IsPolarityOfProjectiveSpace ] );
DeclareOperation( "GeometryOfAbsolutePoints", [ IsPolarityOfProjectiveSpace ] );
DeclareOperation( "AbsolutePoints", [ IsPolarityOfProjectiveSpace ] );
DeclareOperation( "PolarSpace", [ IsPolarityOfProjectiveSpace ] );
[ Dauer der Verarbeitung: 0.18 Sekunden
(vorverarbeitet)
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