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14 Coset Geometries and Diagrams
 14.1 Coset Geometries
 14.2 Diagrams

14 Coset Geometries and Diagrams

This part of FinInG depends on GRAPE.

14.1 Coset Geometries

Suppose we have an incidence geometry Γ (as defined in chapter 4), together with a group G of automorphisms of Γ is also given such that G is transitive on the set of chambers of Γ (also defined in chapter 4). This implies that G is also transitive on the set of all elements of any chosen type i. If we consider a chamber c1,c2,...,cn such that ci is of type i, we can look at the stabilizer Gi of ci in G. The subgroups Gi are called parabolic subgroups of Γ. For a type i, transitivity of G on the elements of type i gives a correspondence between the cosets of the stabilizer Gi and the elements of type i in Γ. Two elements of Γ are incident if and only if the corresponding cosets have a nonempty intersection.

We now use the above observation to define an incidence structure from a group G together with a set of subgroups G1,G2,...,Gn. The type set is {1,2,...,n}. By definition the elements of type i are the (right) cosets of the subgroup Gi. Two cosets are incident if and only if their intersection is not empty. This is an incidence structure which is not necessarily a geometry (see Chapter 4 for definitions).

14.1-1 CosetGeometry
‣ CosetGeometry( G, l )( operation )

Returns: the coset incidence structure 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 incidence structure 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 ] ) )
gap> p:=Random(ElementsOfIncidenceStructure(cg,1));
<element of type 1 of CosetGeometry( SymmetricGroup( [ 1 .. 5 ] ) )>
gap> q:=Random(ElementsOfIncidenceStructure(cg,2));
<element of type 2 of CosetGeometry( SymmetricGroup( [ 1 .. 5 ] ) )>
gap> IsIncident(p,q);
false
gap> IsIncident(p,p);
true
gap> ParabolicSubgroups(cg);
[ Group([ (4,5), (3,5), (1,2)(4,5) ]), Group([ (4,5), (2,3), (1,2,3) ]) ]
gap> Rank(cg) = Size(last);
true
gap> BorelSubgroup(cg);
Group([ (1,2), (4,5) ])
 

14.1-2 IsIncident
‣ IsIncident( ele1, ele2 )( operation )

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

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

14.1-3 ParabolicSubgroups
‣ ParabolicSubgroups( cg )( operation )

Returns: the list of parabolic subgroups defining the coset geometry cg

14.1-4 AmbientGroup
‣ AmbientGroup( cg )( operation )

Returns: the group used to define the coset geometry cg

cg must be a coset geometry.

14.1-5 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 intersection of all parabolic subgrops.

14.1-6 IsFlagTransitiveGeometry
‣ IsFlagTransitiveGeometry( cg )( operation )

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

cg must be a coset geometry.

The group G used to define cg acts naturally on the elements of cg by right translation: a coset G_ig is mapped to G_i(gx) by an element x∈ G. This test is quite time consuming. You can bind the attribute IsFlagTransitiveGeometry if you are sure the coset geometry is indeed flag-transitive.

14.1-7 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.

14.1-8 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.

14.1-9 IsResiduallyConnected
‣ IsResiduallyConnected( cg )( operation )

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

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

This test is quite time consuming. You can bind the attribute IsResiduallyConnected if you are sure the coset geometry is indeed residually connected.

14.1-10 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.

14.1-11 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.

14.1-12 CanonicalResidueOfFlag
‣ CanonicalResidueOfFlag( cg, fl )( operation )

Returns: coset geometry isomorphic to residue of fl in cg

cg must be a coset incidence structure and fl must be a flag in that incidence structure. The returned coset incidence structure for a flag {G_i_1g_i_1,G_i_2g_i_2,... , G_i_kg_i_k} is the coset incidence structure defined by the group H:=∩_j=1^kG_i_j and parabolic subgroups G_j∩ H for j not in the type set {i_1,i_2,... ,i_k} of fl.

14.1-13 ResidueOfFlag
‣ ResidueOfFlag( cg, fl )( operation )

Returns: the residue of fl in cg.

cg must be a coset geometry. CHECK the back-mapping. Still not quite right. I'll have another look.

14.1-14 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! All GRAPE functionality can now be used to analyse cg via its incidence graph.

14.1-15 Rk2GeoGonality
‣ Rk2GeoGonality( cg )( operation )

Returns: the gonality (i.e. half the girth) of the incidence graph of cg.

cg must be a coset geometry of rank 2.

14.1-16 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 (i.e. elements of type 2).

14.1-17 Rk2GeoDiameter
‣ Rk2GeoDiameter( cg, type )( operation )

Returns: the point (or line) diameter.

cg must be a coset geometry of rank 2. type must be either 1 or 2. This function computes the point diameter of cg when type is 1 and the line diameter when type is 2.

14.2 Diagrams

The diagram of a flag-transitive incidence geometry is a schematic description of the structure of the geometry. It is based on the collection of rank 2 residues of the geometry. Since the geometry is flag-transitive, all chambers are equivalent. Let's fix a chamber C = { c1,c2,...,cn}, with ci of type i. For each subset {i,j} of size two in {1,2,...n} we take the residue of the flag C \ { ci, cj}. Flag transitivity ensures that all residues of type I∖{i,j} are isomorphic to each other. For each such residue, the structure is described by some parameters: the gonality and the point and line diameters. For each type i, we also define the i-order to be the elements of type i in the residue of a(ny) flag of type I∖{i}. All this information is depicted in a diagram which is bascically a graph with vertex set I and edges whenever the point diamater, the line diameter and the gonality are all greater than 2.

14.2-1 DiagramOfGeometry
‣ DiagramOfGeometry( Gamma )( operation )

Returns: the diagram of the geometry Gamma

Gamma must be a flag-transitive coset geometry.

The flag-transitivity is not tested by this operation because such test is time consuming. The command IsFlagTransitiveGeometry can be used to check flag-transitivity if needed.

14.2-2 DrawDiagram
‣ DrawDiagram( Diag, filename )( operation )

Returns: does not return anything but writes 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.

In case graphviz is not available on your system, you will get an friendly error message and a file filename.dot will be written. You can then compile this file later or ask a friend to help you.

We illustrate the diagram feature with Neumaier's A8-geometry. The affine space of dimension 3 over the field with two elements is denoted by AG(3,2). If we fix a plane Π in PG(3,2), the structure induced on the 8 points not in Π by the lines and planes of PG(3,2) is isomorphic to AG(3,2). Since every two points of AG(3,2) define a line, the collinearity graph of AG(3,2) (that is the graph whose vertices are the points of AG(3,2) and in which two vertices are adjacent whenever they are collinear) is the complete graph K8 on 8 vertices. Given two copies of the complete graph on 8 vertices, one can label the vertices of each of them with the numbers from 1 to 8. These labelings are always equivalent when the two copies are seen as graphs, but not if they are understood as models of the affine space. The reason is that an affine space has parallel lines and to be affinely equivalent, the labelings must be such that edges which were parallel in the first labeling remain parallel in the second labeling. In fact there are 15 affinely nonequivalent ways to label the vertices of K8. The affine space has 14 planes of 4 points and there are 70 subsets of 4 elements in the vertex set of K8. Each time we label K8, there are 14 of the 70 sets of 4 elements which become planes of AG(3,2). The remaining 4-subsets will be called nonplanes for that labeling. A well-known rank 4 geometry discovered by Neumaier in 1984 can be described using these concepts. This geometry is quite important since its residue of cotype 0 is the famous A7-geometry which is known to be the only flag-transitive locally classical C_3-geometry which is not a polar space (see Aschbacher1984 for details). The Neumaier geometry can be constructed as follows. The elements of types 1 and 2 are the vertices and edges of the complete graph K8, the elements of type 2 are the 4-subsets of the vertex set of K8 and the elements of type 3 are the 15 nonequivalent labelings of K8. Incidences are mostly the natural ones. A 4-subset is incident with a labeling of K8 if it is the set of points of a nonplane in the model of AG(3,2) defined by the labeling.

Alt( [ 1 .. 8 ] )
gap> pabs:= [
>   Group([ (2,4,6), (1,3,2)(4,8)(6,7) ]), 
>   Group([ (1,6,7,8,4), (2,5)(3,4) ]),
>   Group([ (3,6)(7,8), (2,4,5), (1,5)(2,4), (2,4)(6,7), (6,8,7), 
> (1,2)(4,5), (3,7)(6,8) ]),
>   Group([ (1,7,8,4)(2,5,3,6), (1,3)(2,6)(4,8)(5,7), (1,5)(2,4)(3,7)(6,8),
>       (1,8)(2,7)(3,4)(5,6), (1,3)(2,6)(4,7)(5,8) ]) ];
[ Group([ (2,4,6), (1,3,2)(4,8)(6,7) ]), Group([ (1,6,7,8,4), (2,5)(3,4) ]), 
  Group([ (3,6)(7,8), (2,4,5), (1,5)(2,4), (2,4)(6,7), (6,8,7), (1,2)(4,5), 
      (3,7)(6,8) ]), 
  Group([ (1,7,8,4)(2,5,3,6), (1,3)(2,6)(4,8)(5,7), (1,5)(2,4)(3,7)(6,8), 
      (1,8)(2,7)(3,4)(5,6), (1,3)(2,6)(4,7)(5,8) ]) ]
gap> cg:=CosetGeometry(g,pabs);
CosetGeometry( AlternatingGroup( [ 1 .. 8 ] ) )
gap> diag:=DiagramOfGeometry(cg);
< Diagram of CosetGeometry( AlternatingGroup( [ 1 .. 8 ] ) , 
[ Group( [ (2,4,6), (1,3,2)(4,8)(6,7) ] ), 
  Group( [ (1,6,7,8,4), (2,5)(3,4) ] ), 
  Group( [ (3,6)(7,8), (2,4,5), (1,5)(2,4), (2,4)(6,7), (6,8,7), (1,2)(4,5), 
      (3,7)(6,8) ] ), 
  Group( [ (1,7,8,4)(2,5,3,6), (1,3)(2,6)(4,8)(5,7), (1,5)(2,4)(3,7)(6,8), 
      (1,8)(2,7)(3,4)(5,6), (1,3)(2,6)(4,7)(5,8) ] ) ] ) >
gap> DrawDiagram(diag, "neuma8");
gap> #Exec("gv neuma8.ps");
gap> point:=Random(ElementsOfIncidenceStructure(cg,1));
<element of type 1 of CosetGeometry( AlternatingGroup( [ 1 .. 8 ] ) )>
gap> residue:=ResidueOfFlag(cg,[point]);
CosetGeometry( Group( [ (1,3,5), (1,7,2)(3,8)(5,6) ] ) )
gap> diagc3:=DiagramOfGeometry(residue);
< Diagram of CosetGeometry( Group( [ (1,3,5), (1,7,2)(3,8)(5,6) ] ) , 
[ Group( [ (2,3,8), (2,3,6), (5,8,6), (2,7,5,6,8) ] ), 
  Group( [ (2,5)(6,8), (1,7,3), (1,7)(5,6), (5,8,6), (2,6)(5,8) ] ), 
  Group( [ (1,6,3,2)(5,8), (2,7)(3,8) ] ) ] ) >
gap> DrawDiagram(diagc3, "a7geo");
gap> #Exec("gv a7geo.ps");
 

The produced diagrams are included here: Neumaier's A8

The A7 geometry:

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