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GAP 4 Package FinInG

Finite Incidence Geometry

1.01

2012

John Bamberg, Anton Betten, Philippe Cara, Jan De Beule, Michel Lavrauw, Max Neunhoeffer.

Copyright

© 2012 by the authors

This package may be distributed under the terms and conditions of the GNU Public License Version 2 or higher.

Acknowledgements

The development phase of FinInG started roughly in 2005. The idea to write a package for projective geometry in GAP was known before, and resulted in pg, a relic that still can be found in the undeposited packages section of the GAP website. One of the main objectives was to develop the new package was to create a tighter connection between finite geometry and the group theoretical functions present in GAP.

The authors thank Michael Pauley, Maska Law and Sven Reichard, for their contributions during the early days of the project.

Contents

1.2 Overview over this manual

1.3 The Development Team

2 Installation of the FinInG-Package

2.1 Installing FinInG under UNIX like systems

2.2 Installing other required packages

3.1 A simple example to get you started

3.2 Polar Spaces

  3.2-1 Lines meeting a hermitian curve
  3.2-2 W(3,3) inside W(5,3)
  3.2-3 Spreads of W(5,3)
  3.2-4 The Patterson ovoid

3.3 Elation generalised quadrangles

3.3-1 The classical q-clan
3.3-2 Two ways to construct a flock generalised quadrangle from a Kantor-Knuth semifield q-clan

3.4 Diagram geometries

3.4-1 A rank 4 geometry for PSL(2,11)

4 Incidence Geometry

4.1 Incidence structures

  4.1-1 IsIncidenceStructure
  4.1-2 IncidenceStructure
  4.1-3 IsIncidenceGeometry
  4.1-4 Main categories in IsIncidenceGeometry
  4.1-5 Main categories in IsLieGeometry
  4.1-6 Categories in IsGeneralisedPolygon
  4.1-7 IsElationGQ
  4.1-8 IsClassicalGQ
  4.1-9 Examples of categories of incidence geometries
  4.1-10 TypesOfElementsOfIncidenceStructure
  4.1-11 Rank

4.2 Elements of incidence structures

  4.2-1 Main categories for individual elements of incidence structures
  4.2-2 Main categories for collections of elements of incidence structures
  4.2-3 ElementsOfIncidenceStructure
  4.2-4 Short names for ElementsOfIncidenceStructure
  4.2-5 IsIncident
  4.2-6 Random
  4.2-7 AmbientGeometry

4.3 Flags of incidence structures

4.3-1 FlagOfIcidenceStructure
4.3-2 IsChamberOfIncidenceStructure

4.4 Shadow of elements

  4.4-1 ShadowOfElement
  4.4-2 ShadowOfFlag
  4.4-3 ElementsIncidentWithElementOfIncidenceStructure
  4.4-4 Short names for ElementsIncidentWithElementOfIncidenceStructure

4.5 Enumerating elements of an incidence structure

4.5-1 Iterator
4.5-2 Enumerator

4.6 Lie geometries

  4.6-1 UnderlyingVectorSpace
  4.6-2 ProjectiveDimension
  4.6-3 AmbientSpace
  4.6-4 IsEmptySubspace

4.7 Elements of Lie geometries

  4.7-1 VectorSpaceToElement
  4.7-2 ElementToVectorSpace
  4.7-3 \in
  4.7-4 More short names for ElementsIncidentWithElementOfIncidenceStructure

4.8 Hard wired embeddings and converting elements

4.8-1 ElementToElement

5 Projective Spaces

5.1 Projective Spaces and basic operations

  5.1-1 IsProjectiveSpace
  5.1-2 ProjectiveSpace
  5.1-3 ProjectiveDimension
  5.1-4 BaseField
  5.1-5 UnderlyingVectorSpace
  5.1-6 AmbientSpace

5.3 Shadows of Projective Subspaces

  5.3-1 ShadowOfElement
  5.3-2 ShadowOfFlag
  5.3-3 ElementsIncidentWithElementOfIncidenceStructure
  5.3-4 Short names for ElementsIncidentWithElementOfIncidenceStructure

5.4 Enumerating subspaces of a projective spaces

  5.4-1 Iterator
  5.4-2 Enumerator
  5.4-3 List

6 Projective Groups

6.1 Projectivities, collineations and correlations of projective spaces.

  6.1-1 Categories for group elements
  6.1-2 Representations for group elements
  6.1-3 Projectivities
  6.1-4 Collineations of projective spaces
  6.1-5 Projective strictly semilinear maps
  6.1-6 Correlations and collineations

6.2 Construction of projectivities, collineations and correlations.

  6.2-1 Projectivity
  6.2-2 CollineationOfProjectiveSpace
  6.2-3 ProjectiveSemilinearMap
  6.2-4 IdentityMappingOfElementsOfProjectiveSpace
  6.2-5 StandardDualityOfProjectiveSpace
  6.2-6 CorrelationOfProjectiveSpace

6.3 Basic operations for projectivities, collineations and correlations of projective spaces

  6.3-1 Representative
  6.3-2 UnderlyingMatrix
  6.3-3 BaseField
  6.3-4 FieldAutomorphism
  6.3-5 ProjectiveSpaceIsomorphism
  6.3-6 Order

6.4 The groups PΓL, PGL, and PSL in FinInG

  6.4-1 ProjectivityGroup
  6.4-2 CollineationGroup
  6.4-3 SpecialProjectivityGroup
  6.4-4 IsProjectivityGroup
  6.4-5 IsCollineationGroup

6.5 Basic operations for projective groups

6.5-1 BaseField
6.5-2 Dimension

6.6 Natural embedding of a collineation group in a correlation group

6.6-1 Embedding

6.7 Basic action of projective group element

6.8 Projective group actions

  6.8-1 OnProjSubspaces
  6.8-2 ActionOnAllProjPoints
  6.8-3 OnProjSubspacesExtended

6.9 Special subgroups of the projectivity group

  6.9-1 ElationOfProjectiveSpace
  6.9-2 ProjectiveElationGroup
  6.9-3 HomologyOfProjectiveSpace
  6.9-4 ProjectiveHomologyGroup

6.10 Nice Monomorphisms

  6.10-1 NiceMonomorphism
  6.10-2 NiceObject
  6.10-3 CanComputeActionOnPoints

7 Polarities of Projective Spaces

7.1 Creating polarities of projective spaces

  7.1-1 PolarityOfProjectiveSpace
  7.1-2 PolarityOfProjectiveSpace
  7.1-3 PolarityOfProjectiveSpace
  7.1-4 PolarityOfProjectiveSpace

7.2 Operations, attributes and properties for polarties of projective spaces

  7.2-1 SesquilinearForm
  7.2-2 BaseField
  7.2-3 GramMatrix
  7.2-4 CompanionAutomorphism
  7.2-5 IsHermitianPolarityOfProjectiveSpace
  7.2-6 IsSymplecticPolarityOfProjectiveSpace
  7.2-7 IsOrthogonalPolarityOfProjectiveSpace
  7.2-8 IsPseudoPolarityOfProjectiveSpace

7.3 Polarities, absolute points, totally isotropic elements and finite classical polar spaces

  7.3-1 GeometryOfAbsolutePoints
  7.3-2 AbsolutePoints
  7.3-3 PolarSpace

7.4 Commuting polarities

8 Finite Classical Polar Spaces

8.1 Finite Classical Polar Spaces

8.1-1 IsClassicalPolarSpace
8.1-2 PolarSpace

8.2 Canonical and standard Polar Spaces

  8.2-1 SymplecticSpace
  8.2-2 HermitianPolarSpace
  8.2-3 ParabolicQuadric
  8.2-4 HyperbolicQuadric
  8.2-5 EllipticQuadric
  8.2-6 CanonicalPolarSpace
  8.2-7 StandardPolarSpace

8.3 Basic operations for finite classical polar spaces

  8.3-1 UnderlyingVectorSpace
  8.3-2 AmbientSpace
  8.3-3 ProjectiveDimension
  8.3-4 Rank
  8.3-5 BaseField
  8.3-6 IsHyperbolicQuadric
  8.3-7 IsEllipticQuadric
  8.3-8 IsParabolicQuadric

8.4 Subspaces of finite classical polar spaces

  8.4-1 VectorSpaceToElement
  8.4-2 EmptySubspace
  8.4-3 ProjectiveDimension
  8.4-4 ElmentsOfIncidenceStructure
  8.4-5 AmbientSpace
  8.4-6 Coordinates
  8.4-7 Incidence and containment
  8.4-8 Span
  8.4-9 Meet
  8.4-10 IsCollinear
  8.4-11 Polarity
  8.4-12 TypeOfSubspace

8.5 Projective Orthogonal/Unitary/Symplectic groups in FinInG

  8.5-1 SpecialIsometryGroup
  8.5-2 IsometryGroup
  8.5-3 SimilarityGroup
  8.5-4 CollineationGroup

8.6 Enumerating subspaces of polar spaces

  8.6-1 Enumerators for polar spaces
  8.6-2 Enumerator
  8.6-3 Iterators for polar spaces
  8.6-4 Iterator
  8.6-5 AsList

9 Actions, stabilisers and orbits

9.1 Stabilisers

  9.1-1 FiningStabiliser
  9.1-2 FiningStabiliserOrb
  9.1-3 FiningSetwiseStabiliser

10 Affine Spaces

10.1 Affine spaces and basic operations

  10.1-1 IsAffineSpace
  10.1-2 AffineSpace
  10.1-3 Dimension
  10.1-4 BaseField
  10.1-5 UnderlyingVectorSpace
  10.1-6 AmbientSpace

10.2 Subspaces of affine spaces

  10.2-1 AffineSubspace
  10.2-2 ElementsOfIncidenceStructure
  10.2-3 Short names for ElementsOfIncidenceStructure
  10.2-4 Incidence and containment
  10.2-5 AmbientSpace
  10.2-6 BaseField
  10.2-7 Span
  10.2-8 Meet
  10.2-9 IsParallel
  10.2-10 ParallelClass

10.3 Shadows of Affine Subspaces

10.3-1 ShadowOfElement
10.3-2 ShadowOfFlag

10.4 Iterators and enumerators

  10.4-1 Iterator
  10.4-2 Enumerator
  10.4-3 IsVectorSpaceTransversal
  10.4-4 VectorSpaceTransversal
  10.4-5 VectorSpaceTransversalElement
  10.4-6 ComplementSpace

10.5 Affine groups

  10.5-1 AffineGroup
  10.5-2 CollineationGroup
  10.5-3 OnAffineSpaces

11 Geometry Morphisms

11.1 Geometry morphisms in FinInG

11.1-1 IsGeometryMorphism
11.1-2 Intertwiner

11.2 Isomorphisms between polar spaces

11.2-1 IsomorphismPolarSpaces

11.3 When will you use geometry morphisms?

11.4 Natural geometry morphisms

  11.4-1 NaturalEmbeddingBySubspace
  11.4-2 NaturalEmbeddingByFieldReduction
  11.4-3 Intertwiner
  11.4-4 NaturalEmbeddingByFieldReduction
  11.4-5 BlownUpProjectiveSpace
  11.4-6 BlownUpProjectiveSpaceBySubfield
  11.4-7 BlownUpSubspaceOfProjectiveSpace
  11.4-8 BlownUpSubspaceOfProjectiveSpaceBySubfield
  11.4-9 IsDesarguesianSpreadElement
  11.4-10 NaturalEmbeddingBySubField
  11.4-11 NaturalProjectionBySubspace

11.5 Some special kinds of geometry morphisms

  11.5-1 KleinCorrespondence
  11.5-2 NaturalDuality
  11.5-3 ProjectiveCompletion

12 Algebraic Varieties

12.1 Algebraic Varieties

  12.1-1 AlgebraicVariety
  12.1-2 DefiningListOfPolynomials
  12.1-3 AmbientSpace
  12.1-4 PointsOfAlgebraicVariety
  12.1-5 Iterator
  12.1-6 \in

12.2 Projective Varieties

12.2-1 ProjectiveVariety

12.3 Quadrics and Hermitian varieties

  12.3-1 HermitianVariety
  12.3-2 HermitianVariety
  12.3-3 QuadraticVariety
  12.3-4 QuadraticForm
  12.3-5 SesquilinearForm
  12.3-6 PolarSpace

12.4 Affine Varieties

12.4-1 AffineVariety

12.5 Geometry maps

  12.5-1 Source
  12.5-2 Range
  12.5-3 ImageElm
  12.5-4 ImagesSet
  12.5-5 \^

12.6 Segre Varieties

  12.6-1 SegreVariety
  12.6-2 PointsOfSegreVariety
  12.6-3 SegreMap
  12.6-4 Source

12.7 Veronese Varieties

  12.7-1 VeroneseVariety
  12.7-2 PointsOfVeroneseVariety
  12.7-3 VeroneseMap
  12.7-4 Source

12.8 Grassmann Varieties

  12.8-1 GrassmannVariety
  12.8-2 PointsOfGrassmannVariety
  12.8-3 GrassmannMap
  12.8-4 Source

13 Generalised Polygons

13.1 Projective planes

13.1-1 ProjectivePlaneByBlocks
13.1-2 ProjectivePlaneByIncidenceMatrix

13.2 Generalised quadrangles

  13.2-1 IsKantorFamily
  13.2-2 EGQByKantorFamily
  13.2-3 IsqClan
  13.2-4 qClan
  13.2-5 EGQByqClan
  13.2-6 KantorFamilyByqClan
  13.2-7 Particular q-Clans
  13.2-8 BLTSetByqClan
  13.2-9 EGQByBLTSet
  13.2-10 ElationGroup
  13.2-11 BasePointOfEGQ

13.3 Generalised hexagons

  13.3-1 SplitCayleyHexagon
  13.3-2 TwistedTrialityHexagon
  13.3-3 AmbientSpace and AmbientPolarSpace

13.4 General attributes and operations for generalised polygons

  13.4-1 Order
  13.4-2 AmbientSpace
  13.4-3 CollineationGroup
  13.4-4 CollineationAction
  13.4-5 BlockDesignOfGeneralisedPolygon
  13.4-6 IncidenceGraphOfGeneralisedPolygon
  13.4-7 IncidenceMatrixOfGeneralisedPolygon

A The structure of FinInG

A.1 The different components

A.2 The complete inventory

A.2-1 Declarations
A.2-2 Functions/Methods

A.3 The filter graph(s)

B The finite classical groups in FinInG

B.1 Standard forms used to produce the finite classical groups.

B.1-1 CanonicalGramMatrix
B.1-2 CanonicalQuadraticForm

B.2 Direct commands to construct the projective classical groups in FinInG

  B.2-1 SOdesargues
  B.2-2 GOdesargues
  B.2-3 SUdesargues
  B.2-4 GUdesargues
  B.2-5 Spdesargues
  B.2-6 GeneralSymplecticGroup
  B.2-7 GSpdesargues
  B.2-8 GammaSp
  B.2-9 DeltaOminus
  B.2-10 DeltaOplus
  B.2-11 GammaOminus
  B.2-12 GammaO
  B.2-13 GammaOplus
  B.2-14 GammaU

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