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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

13 Coset Geometries and Diagrams

13.1 Preliminary version of the manual

Suppose we have a flag-transitive incidence geometry Gamma. This means that a group G of automorphisms of Gamma is also given such that G is transitive on the set of chambers of Gamma. This implies that G is also transitive on the set of all elements of any chosen type i. If we consider a chamber c_1,c_2,... ,c_n such that c_i is of type i, we can look at the stabilizer G_i of c_i in G. The subgroups G_i are called parabolic subgroups of Gamma. For a type i, transitivity of G on the elements of type i gives a correspondence between the cosets of the stabilizer G_i and the elements of type i in Gamma. Two elements of Gamma are incident if and only if the corresponding cosets have a nonempty intersection. We now use the above oservation to define an incidence structure from a group G together with a set of subgroups G_1,G_2,... ,G_n. The type set is {1,2,...,n}. By definition the elements of type i are the (right) cosets of the subgroup G_i. Two cosets are incident if and only if their intersection is not empty.

13.1-1 CosetGeometry
> CosetGeometry( G, l )( operation )

Returns: Returns the coset geometry defined by the list l of subgroups of the group G

G must be a group and l is a list of subgroups of G. The subgroups in l will be the parabolic subgroups of the coset geometry whose rank equals the length of l.

gap> g:=SymmetricGroup(5);
Sym( [ 1 .. 5 ] )
gap> g1:=Stabilizer(g,[1,2],OnSets);
Group([ (4,5), (3,5), (1,2)(4,5) ])
gap> g2:=Stabilizer(g,[1,2,3],OnSets);
Group([ (4,5), (2,3), (1,2,3) ])
gap> cg:=CosetGeometry(g,[g1,g2]);
CosetGeometry( SymmetricGroup( [ 1 .. 5 ] ) )
 

13.1-2 IsIncident
> IsIncident( ele1, ele2 )( operation )

Returns: true if ans only if ele1 and ele2 are incident

ele1 and ele2 must be two elements in thes same coset geometry.

13.1-3 ParabolicSubgroups
> ParabolicSubgroups( cg )( operation )

Returns: List of parabolic subgroups defining the coset geometry cg

13.1-4 AmbientGroup
> AmbientGroup( cg )( operation )

Returns: the group of the coset geometry cg

cg must be a coset geometry.

13.1-5 DiagramOfGeometry
> DiagramOfGeometry( Gamma )( operation )

Returns: The diagram of the geometry Gamma

Gamma must be a coset geometry.

13.1-6 DrawDiagram
> DrawDiagram( Diag, filename )( operation )

Returns: Does not return anything but wirtes a file filename.ps

Diag must be a diagram. Writes a file filename.ps in the current directory with a pictorial version of the diagram. This command uses the graphviz package which is available from http://www.graphviz.org.

13.1-7 Borelsubgroup
> Borelsubgroup( cg )( operation )

Returns: The Borel subgroup of de geometry cg

The Borel subgroup is equal to the stabilizer of a chamber. It corresponds to the untersection of all parabolic subgrops.

13.1-8 IsFlagTransitiveGeometry
> IsFlagTransitiveGeometry( cg )( operation )

Returns: true if and only if the group G defining cg acts flag-transitively.

cg must be a coset geometry.

13.1-9 IsFirmGeometry
> IsFirmGeometry( cg )( operation )

Returns: true if and only if cg is firm.

An incidence geometry is said to be firm if every nonmaximal flag is contained in at least two chambers. cg must be a coset geometry.

13.1-10 IsConnected
> IsConnected( cg )( operation )

Returns: true if and only if cg is connected.

A geometry is connected if and only if its incidence graph is connected. cg must be a coset geometry.

13.1-11 IsResiduallyConnected
> IsResiduallyConnected( cg )( operation )

Returns: true if and only if cg is residually connected.

A geometry is residually connected the incidence graphs of all its residues of rank at least 2 are connected. cg must be a coset geometry.

13.1-12 StandardFlagOfCosetGeometry
> StandardFlagOfCosetGeometry( cg )( operation )

Returns: Standard chamber of cg

The standard chamber just consists of all parabolic subgroups (i.e. the trivial cosets of these subgroups). cg must be a coset geometry. Maybe this function should be called StandardChamberOfCosetGeometry.

13.1-13 FlagToStandardFlag
> FlagToStandardFlag( cg, fl )( operation )

Returns: element of the defining group of cg which maps fl to the standard chamber of cg.

fl must be a chamber given as a list of cosets of the parabolic subgroups of cg. The order of the elements of the chamber should be the same as the order of the parabolics defining cg.

13.1-14 CanonicalResidueOfFlag
> CanonicalResidueOfFlag( cg, fl )( operation )

Returns: coset geometry isomorphic to residue of fl in cg

cg must be a coset geometry and fl must be a flag in that geometry. The returned coset geometry for a flag {giGi} is ...

13.1-15 ResidueOfFlag
> ResidueOfFlag( cg, fl )( operation )

Returns: The residue of fl in cg.

CHECK the back-mapping.

13.1-16 IncidenceGraph
> IncidenceGraph( cg )( operation )

Returns: incidence graph of cg.

cg must be a coset geometry. The graph returned is a GRAPE object. Be sure the GRAPE is loaded!

13.1-17 Rank2Parameters
> Rank2Parameters( cg )( operation )

Returns: a list of length 3.

cg must be a coset geometry of rank 2. This function computes the gonality, point and line diameter of cg. These appear as a list in the first entry of the returned list. The second entry contains a list of length 2 with the point order and the total number of points (i.e. elements of type 1) in the geometry. The last entry contains the line order and the number of lines.

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