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GAP 4 Package FinInG

Finite Incidence Geometry

1.01

March 2009

John Bamberg, Anton Betten, Philippe Cara, Jan De Beule, Michel Lavrauw, Maska Law, Max Neunhoeffer, Michael Pauley, Sven Reichard

Copyright

© 2009 by the authors

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

Contents

1.2 Overview over this manual

1.3 The Development Team

2 Installation of the FinInG-Package

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 IsIncidenceGeometry
  4.1-3 Main categories in IsIncidenceGeometry
  4.1-4 Main categories in IsLieGeometry
  4.1-5 Categories in IsGeneralisedPolygon
  4.1-6 IsElationGQ
  4.1-7 IsClassicalGQ
  4.1-8 Examples of categories of incidence geometries
  4.1-9 TypesOfElementsOfIncidenceStructure
  4.1-10 Rank
  4.1-11 AmbientGeometry
  4.1-12 AmbientSpace
  4.1-13 IsIncident

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 Hyperplanes
  4.2-6 IsIncident
  4.2-7 Random
  4.2-8 Size

4.3 Shadows of elements
4.3-1 ShadowOfElement
4.3-2 ShadowOfFlag

5 Projective Spaces

5.1 Creating Projective Spaces and basic operations
  5.1-1 ProjectiveSpace
  5.1-2 ProjectiveDimension
  5.1-3 BaseField
  5.1-4 UnderlyingVectorSpace

5.2 Subspaces of projective spaces
  5.2-1 VectorSpaceToElement
  5.2-2 EmptySubspace
  5.2-3 ProjectiveDimension
  5.2-4 ElmentsOfIncidenceStructure
  5.2-5 StandardFrame
  5.2-6 Coordinates
  5.2-7 EquationOfHyperplane
  5.2-8 AmbientSpace
  5.2-9 BaseField
  5.2-10 AsList
  5.2-11 Random
  5.2-12 RandomSubspace
  5.2-13 Span
  5.2-14 Meet
  5.2-15 IsIncident
  5.2-16 FlagOfIncidenceStructure
  5.2-17 IsEmptyFlag
  5.2-18 IsChamberOfIncidenceStructure

5.3 Shadows of Projective Subspaces
  5.3-1 ShadowOfElement
  5.3-2 ShadowOfFlag
  5.3-3 ShadowOfFlag
  5.3-4 ElementsIncidentWithElementOfIncidenceStructure

6 Projective Groups

6.1 Projectivities, collineations and correlations of projective spaces
  6.1-1 Categories for group elements
  6.1-2 Projectivity
  6.1-3 ProjectiveSemilinearMap
  6.1-4 StandardDualityOfProjectiveSpace
  6.1-5 CorrelationOfProjectiveSpace

6.2 Basic operations for projectivities, collineations and correlations of projective spaces
  6.2-1 Representations for group elements
  6.2-2 UnderlyingMatrix
  6.2-3 BaseField
  6.2-4 FieldAutomorphism
  6.2-5 ProjectiveSpaceIsomorphism

6.3 Collineation groups of projective or polar spaces
  6.3-1 CollineationGroup
  6.3-2 SimilarityGroup
  6.3-3 IsometryGroup
  6.3-4 SpecialIsometryGroup

6.4 Basic operations for projective groups
6.4-1 BaseField
6.4-2 Dimension

6.5 Collineation group as a subgroup of a correlation group

6.6 Projective group actions
  6.6-1 OnProjSubspaces
  6.6-2 ActionOnAllProjPoints
  6.6-3 OnProjSubspacesReverseing

6.7 Nice Monomorphisms
6.7-1 CanComputeActionOnPoints

8 Classical Polar Spaces

8.1 Creating Polar Spaces
8.1-1 PolarSpace

8.2 Canonical Polar Spaces
  8.2-1 SymplecticSpace
  8.2-2 HermitianVariety
  8.2-3 ParabolicQuadric
  8.2-4 HyperbolicQuadric
  8.2-5 EllipticQuadric
  8.2-6 CanonicalPolarSpace

8.3 Basic operations finite classical polar spaces
  8.3-1 UnderlyingVectorSpace
  8.3-2 ProjectiveDimension
  8.3-3 StandardFrame
  8.3-4 Coordinates
  8.3-5 EquationOfHyperplane
  8.3-6 BaseField
  8.3-7 AsList
  8.3-8 Random
  8.3-9 RandomSubspace
  8.3-10 VectorSpaceToElement
  8.3-11 EmptySubspace
  8.3-12 \in
  8.3-13 Span
  8.3-14 Meet
  8.3-15 IsCollinear
  8.3-16 Polarity
  8.3-17 AmbientSpace
  8.3-18 TypeOfSubspace

8.4 All the elements or just one at a time
  8.4-1 Enumerators for polar spaces
  8.4-2 Enumerator
  8.4-3 Iterators for projective spaces
  8.4-4 Iterator

11 Algebraic Varieties

11.1 Projective Varieties
11.1-1 ProjectiveVariety

11.2 Affine Varieties
11.2-1 AffineVariety

13 Coset Geometries and Diagrams

13.1 Preliminary version of the manual
  13.1-1 CosetGeometry
  13.1-2 IsIncident
  13.1-3 ParabolicSubgroups
  13.1-4 AmbientGroup
  13.1-5 DiagramOfGeometry
  13.1-6 DrawDiagram
  13.1-7 Borelsubgroup
  13.1-8 IsFlagTransitiveGeometry
  13.1-9 IsFirmGeometry
  13.1-10 IsConnected
  13.1-11 IsResiduallyConnected
  13.1-12 StandardFlagOfCosetGeometry
  13.1-13 FlagToStandardFlag
  13.1-14 CanonicalResidueOfFlag
  13.1-15 ResidueOfFlag
  13.1-16 IncidenceGraph
  13.1-17 Rank2Parameters

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