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Computer • algebra
Documentation
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Introduction
Construction of Incidence and Coset Geometries
Construction of an Incidence Geometry
IncidenceGeometry(G) : GrphUnd -> IncGeom
Example
IncidenceGeometry_Constructors (H151E1)
Example
IncidenceGeometry_Constructors (H151E2)
Example
IncidenceGeometry_Constructors (H151E3)
Example
IncidenceGeometry_Constructors (H151E4)
Construction of a Coset Geometry
CosetGeometry(G, S, I) : GrpPerm, Set, Set -> CosetGeom
CosetGeometry(G, S) : GrpPerm, Set -> CosetGeom
Example
IncidenceGeometry_Constructors (H151E5)
Example
IncidenceGeometry_Constructors (H151E6)
Example
IncidenceGeometry_Constructors (H151E7)
Elementary Invariants
Points(D) : IncGeom -> SetIndx
Types(D) : IncGeom -> SetIndx
Types(C) : CosetGeom -> SetIndx
Rank(D) : IncGeom -> RngIntElt
Rank(C) : CosetGeom -> RngIntElt
IncidenceGraph(D) : IncGeom -> GrphUnd, GrphVertSet, GrphEdgeSet
Group(C) : CosetGeom -> GrpPerm
MaxParabolics(C) : CosetGeom -> SetIndx
MinParabolics(C) : CosetGeom -> SetIndx
Borel(C) : CosetGeom -> GrpPerm
Kernel(C) : CosetGeom -> GrpPerm
Kernels(C) : CosetGeom -> SeqEnum
Quotient(C, K) : CosetGeom, GrpPerm -> CosetGeom
Conversion Functions
IncidenceGeometry(C) : CosetGeom -> IncGeom
CosetGeometry(D) : IncGeom -> BoolElt, CosetGeom
Graph(D) : IncGeom -> GrphUnd
Graph(C) : CosetGeom -> GrphUnd
Example
IncidenceGeometry_Constructors (H151E8)
Residues
Residue(D, f) : IncGeom, Set -> IncGeom
Residue(C, f) : CosetGeom, Set -> CosetGeom
Truncations
Truncation(D, t) : IncGeom, Set -> IncGeom
Truncation(C, t) : CosetGeom, Set -> CosetGeom
Shadows
Shadow(D, I, F) : IncGeom, Set, Set -> SetIndx
Shadow Spaces
ShadowSpace(D,I) : IncGeom, Set -> Inc
Automorphism Group and Correlation Group
AutomorphismGroup(D) : IncGeom -> GrpPerm
CorrelationGroup(D) : IncGeom -> GrpPerm
Properties of Incidence Geometries and Coset Geometries
IsFTGeometry(D) : IncGeom -> BoolElt
IsFTGeometry(C) : CosetGeom -> BoolElt
IsFirm(X) : IncGeom -> BoolElt
IsThin(X) : CosetGeom -> BoolElt
IsThick(X) : CosetGeom -> BoolElt
IsResiduallyConnected(X) : IncGeom -> BoolElt
IsGraph(D) : IncGeom -> GrphUnd
IsGraph(C) : CosetGeom -> GrphUnd
Intersection Properties of Coset Geometries
HasIntersectionPropertyN(C,n) : CosetGeom, RngIntElt -> BoolElt, BoolElt
HasIntersectionProperty(C) : CosetGeom -> BoolElt
HasWeakIntersectionProperty(C) : CosetGeom -> BoolElt
Primitivity Properties on Coset Geometries
IsPrimitive(C) : CosetGeom -> BoolElt
IsWeaklyPrimitive(C) : CosetGeom -> BoolElt
IsResiduallyPrimitive(C) : CosetGeom -> BoolElt
IsResiduallyWealyPrimitive(C) : CosetGeom -> BoolElt
IsLocallyTwoTransitive(C) : CosetGeom -> BoolElt
LocallySArcTransitive(C) : CosetGeom -> RngIntElt
Diagram of an Incidence Geometry
Diagram(D) : IncGeom -> GrphUnd, GrphVertSet, GrphEdgeSet
Example
IncidenceGeometry_diagram (H151E9)
Example
IncidenceGeometry_diagram (H151E10)
Example
IncidenceGeometry_diagram (H151E11)
Example
IncidenceGeometry_diagram (H151E12)
C-Groups
HasIntersectionProperty(G) : GrpPerm -> BoolElt
HasStringProperty(G) : GrpPerm -> BoolElt
IsCGroup(G) : GrpPerm -> BoolElt
IsStringCGroup(G) : GrpPerm -> BoolElt
CosetGeometryFromCGroup(G) : GrpPerm -> CosetGeom
CosetGeometryToCGroup(D) : CosetGeom -> GrpPerm
CoxeterDiagram(G) : GrpPerm -> GrphUnd
Example
IncidenceGeometry_cosetgeometrytocgroup (H151E13)
Example
IncidenceGeometry_cgrouptocosetgeometry (H151E14)
C^ +-Groups
HasIntersectionPropertyPlus(G) : GrpPerm -> BoolElt
CosetGeometryFromCPlusGroup(G) : GrpPerm -> CosetGeom
BDiagram(G) : GrpPerm -> GrphUnd
Example
IncidenceGeometry_cgrouptocosetgeometry (H151E15)
Bibliography
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V2.28, 13 July 2023