<!-- 
diagramgeometries.xml    FinInG package documentation
                                                                   John Bamberg
                                                                   Anton Betten
                                                                  Philippe Cara
                                                                   Jan De Beule
                                                                 Michel Lavrauw
                                                                Max Neunhoeffer

  Copyright (C) 2017,	Colorado State University
                        Sabancı Üniversitesi
						Università degli Studi di Padova
						Universiteit Gent
						University of St. Andrews
						University of Western Australia
                        Vrije Universiteit Brussel

This is the chapter of the documentation describing diagram geometries groups.

-->

<Chapter Label="diagramgeometries">
<Heading>Coset Geometries and Diagrams</Heading>

This part of <Package>FinInG</Package> depends
on <Package>GRAPE</Package>. 

<Section>
<Heading>Coset Geometries</Heading>

Suppose we have an <E>incidence
geometry</E> &Gamma; (as defined in chapter <Ref Chap="incidencegeometry"/>), together with
a group <M>G</M> of automorphisms of &Gamma; such that <M>G</M> is
transitive on the set of <E>chambers</E> of &Gamma; (also
defined in chapter <Ref Chap="incidencegeometry"/>). This implies
that <M>G</M> is also transitive on the set of all elements of any
chosen type <M>i</M>. If we consider a chamber <Alt Not="HTML"><M>\{c_{1},c_{2},... ,c_{n}\}</M></Alt>
<Alt Only="HTML MathJax"><M>\{c_{1},c_{2},... ,c_{n}\}</M></Alt>
<Alt Only="HTML noMathJax">c&lt;sub>1&lt;/sub>,c&lt;sub>2&lt;/sub>,...,c&lt;sub>n&lt;/sub></Alt>
such that <Alt Not="HTML"><M>c_{i}</M></Alt><Alt Only="HTML MathJax"><M>c_{i}</M></Alt><Alt Only="HTML noMathJax">c&lt;sub>i&lt;/sub></Alt> is of type <M>i</M>,
we can look at the stabilizer <Alt Not="HTML"><M>G_{i}</M></Alt><Alt Only="HTML MathJax"><M>G_{i}</M></Alt><Alt Only="HTML noMathJax">G&lt;sub>i&lt;/sub></Alt>
of <Alt Not="HTML"><M>c_{i}</M></Alt><Alt Only="HTML MathJax"><M>c_{i}</M></Alt><Alt Only="HTML noMathJax">c&lt;sub>i&lt;/sub></Alt> in <M>G</M>. The
subgroups  <Alt Not="HTML"><M>G_{i}</M></Alt><Alt Only="HTML MathJax"><M>G_{i}</M></Alt><Alt Only="HTML noMathJax">G&lt;sub>i&lt;/sub></Alt> are
called <E>parabolic subgroups</E> of &Gamma;. For a
type <M>i</M>, transitivity of <M>G</M> on the elements of
type <M>i</M> gives a correspondence between the cosets of the
stabilizer <Alt Not="HTML"><M>G_{i}</M></Alt><Alt Only="HTML MathJax"><M>G_{i}</M></Alt>
<Alt Only="HTML noMathJax">G&lt;sub>i&lt;/sub></Alt> and the elements of type <M>i</M>
in &Gamma;. Two elements of &Gamma; are incident if and only
if the corresponding cosets have a nonempty intersection.<P/>

We now use the above observation to define an incidence structure from
a group <M>G</M> together with a set of
subgroups <Alt Not="HTML"><M>\{G_{1},G_{2},...,G_{n}\}</M></Alt><Alt Only="HTML MathJax"><M>\{G_{1},G_{2},...,G_{n}\}</M></Alt>
<Alt Only="HTML noMathJax">G&lt;sub>1&lt;/sub>,G&lt;sub>2&lt;/sub>,...,G&lt;sub>n&lt;/sub></Alt>.
The type set is <Alt Not="HTML"><M>\{1,2,...,n\}</M></Alt><Alt Only="HTML MathJax"><M>\{1,2,...,n\}</M></Alt><Alt Only="HTML noMathJax">&#123;1,2,...,n&#125;</Alt>.
By definition the elements of type <M>i</M> are the (right) cosets of the subgroup 
<Alt Not="HTML"><M>G_{i}</M></Alt><Alt Only="HTML MathJax"><M>G_{i}</M></Alt><Alt Only="HTML noMathJax">G&lt;sub>i&lt;/sub></Alt>. 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 <Ref Chap="incidencegeometry"/> for definitions). In order to check whether a coset incidence structure is indeed a
geometry you can use the command <F>IsFlagTransitiveGeometry</F> which (in case it returns <F>true</F>) guarantees that the argument is a geometry.

<ManSection>
<Filt Name="IsCosetGeometry" Type="Category"/>
<Description>
This category is a subcategory of <C>IsIncidenceGeometry</C>, and contains all coset geometries.
</Description>
</ManSection>

<ManSection>
<Oper Name="CosetGeometry" Arg="G,l"/>
<Returns>the coset incidence structure defined by the list <A>l</A> of subgroups of the group <A>G</A>.</Returns>
<Description> 
<A>G</A> must be a group and <A>l</A> is a list of subgroups
of <A>G</A>. The subgroups in <A>l</A> will be the <E>parabolic subgroups</E>
of the <E>coset incidence structure</E> whose rank equals the length
of <A>l</A>. 
<Example>
<#Include SYSTEM "../examples/include/diagram_cosetgeom.include"> 
</Example>
</Description>
</ManSection>

<ManSection>
<Oper Name="IsIncident" Arg="ele1,ele2"/>
<Returns>true if and only if <A>ele1</A> and <A>ele2</A> are incident</Returns>
<Description> 
<A>ele1</A> and <A>ele2</A> must be two elements in the same coset geometry.
</Description>
</ManSection>

<ManSection>
<Oper Name="ParabolicSubgroups" Arg="cg"/>
<Returns>the list of parabolic subgroups defining the coset geometry <A>cg</A> </Returns>
<Description> 
</Description>
</ManSection>

<ManSection>
<Oper Name="AmbientGroup" Arg="cg"/>
<Returns>the group used to define the coset geometry <A>cg</A></Returns>
<Description> 
<A>cg</A> must be a coset geometry.
</Description>
</ManSection>

<ManSection>
<Oper Name="Borelsubgroup" Arg="cg"/>
<Returns>the Borel subgroup of de geometry <A>cg</A></Returns>
<Description> 
The Borel subgroup is equal to the stabilizer of a chamber. It corresponds to the intersection of all parabolic subgroups.
</Description>

</ManSection>
<ManSection>
<Oper Name="RandomElement" Arg="cg"/>
<Returns>a random element of <A>cg</A></Returns>
<Description> 
<A>cg</A> must be a coset geometry.
</Description>
</ManSection>

<ManSection>
<Oper Name="RandomFlag" Arg="cg"/>
<Returns>a random flag of <A>cg</A></Returns>
<Description> 
<A>cg</A> must be a coset geometry.
</Description>
</ManSection>

<ManSection>
<Oper Name="RandomChamber" Arg="cg"/>
<Returns>a random chamber of <A>cg</A></Returns>
<Description> 
<A>cg</A> must be a coset geometry.
</Description>
</ManSection>

<Example>
<#Include SYSTEM "../examples/include/diagram_random.include"> 
</Example>

<ManSection>
<Oper Name="IsFlagTransitiveGeometry" Arg="cg"/>
<Returns>true if and only if the group <A>G</A> defining <A>cg</A>
  acts flag-transitively.</Returns>  
<Description> 
<A>cg</A> must be a coset geometry.<P/>

The group <M>G</M> used to define <A>cg</A> acts naturally on the
elements of <A>cg</A> by right translation: a coset <M>G_ig</M> is
mapped to <M>G_i(gx)</M> by an element <M>x\in G</M>.

This test can be quite time consuming. You can bind the
attribute <C>IsFlagTransitiveGeometry</C> if you are sure the coset geometry is
indeed flag-transitive.
</Description>
</ManSection>

<Example>
<#Include SYSTEM "../examples/include/diagram_notFT.include"> 
</Example>

<ManSection>
<Oper Name="OnCosetGeometryElement" Arg="ele,g"/>
<Returns>the image of the <F>CosetGeometryElement</F> <A>ele</A> under the action of <A>g</A></Returns>
<Description> 
The group element <A>g</A> must belong to <F>AmbientGroup(AmbientGeometry(ele))</F>. 
</Description>
</ManSection>

<ManSection>
    <Oper Name="\^" Arg="ele,g"/>
 <Returns>an element of a coset geometry</Returns>
<Description>
This is an operation which returns the image of <A>ele</A>, an element of a coset incidence structure, under <A>g</A>, an element of <F>AmbientGroup(AmbientGeometry(ele))</F>.
 </Description>
</ManSection>

<ManSection>
    <Oper Name="\^" Arg="flg,g"/>
 <Returns>a flag of a coset geometry</Returns>
<Description>
This is an operation which returns the image of <A>flg</A>, a flag of a coset incidence structure, under <A>g</A>, an element of <F>AmbientGroup(AmbientGeometry(flg))</F>.
 </Description>
</ManSection>

<ManSection>
<Oper Name="IsFirmGeometry" Arg="cg"/>
<Returns>true if and only if <A>cg</A> is firm.</Returns>
<Description> 
An incidence geometry is said to be <E>firm</E> if every nonmaximal flag is contained in at least two chambers. <A>cg</A> must be a coset geometry. 
</Description>
</ManSection>

<ManSection>
<Oper Name="IsThickGeometry" Arg="cg"/>
<Returns>true if and only if <A>cg</A> is thick.</Returns>
<Description> 
An incidence geometry is said to be <E>thick</E> if every nonmaximal flag is contained in at least three chambers. <A>cg</A> must be a coset geometry. 
</Description>
</ManSection>

<ManSection>
<Oper Name="IsThinGeometry" Arg="cg"/>
<Returns>true if and only if <A>cg</A> is thin.</Returns>
<Description> 
An incidence geometry is said to be <E>thin</E> if every rank one residue contains exactly 2 elements.
This means that every comaximal flag is contained in exactly 2 chambers. <A>cg</A> must be a coset geometry.
</Description>
</ManSection>

<Example>
<#Include SYSTEM "../examples/include/diagram_firmthinthick.include"> 
</Example>

<ManSection>
<Oper Name="IsConnected" Arg="cg"/>
<Returns>true if and only if <A>cg</A> is connected.</Returns>
<Description> 
A geometry is <E>connected</E> if and only if its incidence graph is connected. <A>cg</A> must be a coset geometry.
</Description>
</ManSection>

<ManSection>
<Oper Name="IsResiduallyConnected" Arg="cg"/>
<Returns>true if and only if <A>cg</A> is residually connected.</Returns>
<Description> 
A geometry is <E>residually connected</E> if the incidence graphs of all its
residues of rank at least 2 are connected. <A>cg</A> must be a coset
geometry.<P/> 

This test is quite time consuming. You can bind the
attribute <C>IsResiduallyConnected</C> if you are sure the coset geometry is
indeed residually connected.
</Description>
</ManSection>

<Example>
<#Include SYSTEM "../examples/include/diagram_connectedness.include"> 
</Example>

<ManSection>
<Oper Name="StandardFlagOfCosetGeometry" Arg="cg"/>
<Returns>standard chamber of <A>cg</A></Returns>
<Description> 
The standard chamber just consists of all parabolic subgroups
(i.e. the trivial cosets of these subgroups). The object returned is a FlagOfIncidenceStructure. <A>cg</A> must be a
coset geometry. 
<!--Maybe this function should be called StandardChamberOfCosetGeometry.-->
</Description>
</ManSection>

<ManSection>
<Oper Name="FlagToStandardFlag" Arg="cg,fl"/>
<Returns>element of the defining group of <A>cg</A> which maps <A>fl</A> to the standard chamber of <A>cg</A>.</Returns>
<Description> 
<A>fl</A> must be a chamber given as a list of cosets of the parabolic subgroups of <A>cg</A>.
</Description>
</ManSection>

<Example>
<#Include SYSTEM "../examples/include/diagram_flagmapping.include"> 
</Example>


<ManSection>
<Oper Name="CanonicalResidueOfFlag" Arg="cg,fl"/>
<Returns>coset geometry isomorphic to residue of <A>fl</A> in <A>cg</A></Returns>
<Description> 
<A>cg</A> must be a coset incidence structure and <A>fl</A> must be a flag in
that incidence structure. The returned coset incidence structure for a
flag <M>\{G_{i_1}g_{i_1},G_{i_2}g_{i_2},\ldots , G_{i_k}g_{i_k}\}</M>
is the coset incidence structure defined by the
group <M>H:=\cap_{j=1}^{k}G_{i_j}</M> and parabolic
subgroups <M>G_{j}\cap H</M> for <M>j</M> not in the type
set <M>\{i_1,i_2,\ldots ,i_k\}</M> of <A>fl</A>.
</Description>
</ManSection>

<ManSection>
<Oper Name="ResidueOfFlag" Arg="fl"/>
<Returns>the residue of <A>fl</A> in <F>AmbientGeometry(fl)</F>.</Returns>
<Description> 
This is a <F>CosetGeometry</F> method for the <F>ResidueOfFlag</F> operation given in Chapter <Ref Chap="incidencegeometry"/>.
Note that the related operation <F>CanonicalResidueOfFlag</F> takes <E>two</E> arguments.
</Description>
</ManSection>

<Example>
<#Include SYSTEM "../examples/include/diagram_residues.include"> 
</Example>

<ManSection>
<Oper Name="IncidenceGraph" Arg="cg"/>
<Returns>incidence graph of <A>cg</A>.</Returns>
<Description> 
<A>cg</A> must be a coset geometry. The graph returned is
a <Package>GRAPE</Package> object. All <Package>GRAPE</Package>
functionality can now be used to analyse <A>cg</A> via its incidence graph.
</Description>
</ManSection>

<ManSection>
<Oper Name="Rk2GeoGonality" Arg="cg"/>
<Returns>the gonality (i.e. half the girth) of the incidence graph of <A>cg</A>.</Returns>
<Description> 
<A>cg</A> must be a coset geometry of rank 2.
</Description>
</ManSection>

<ManSection>
<Oper Name="Rk2GeoDiameter" Arg="cg,type"/>
<Returns>the point (or line) diameter.</Returns>
<Description> 
<A>cg</A> must be a coset geometry of rank 2. <A>type</A> must be
either <C>1</C> or <C>2</C>. This function computes
the point diameter of <A>cg</A> when <A>type</A> is <C>1</C> and the
line diameter when <A>type</A> is <C>2</C>.
</Description>
</ManSection>

<ManSection>
<Oper Name="GeometryOfRank2Residue" Arg="resi"/>
<Returns>the geometry of the <F>Rank2Residue</F> object <A>resi</A>.</Returns>
<Description> 
The rank 2 residues of a geometry are fundamental when dealing with diagrams. Therefore they 
are kept in an attribute as (a list of) objects of type <F>Rank2Residue</F>. The present operation just extracts
the residue as a coset geometry from such a <F>Rank2Residue</F> object.
</Description>
</ManSection>

<ManSection>
<Oper Name="Rank2Parameters" Arg="cg"/>
<Returns>a list of length 3.</Returns>
<Description> 
<A>cg</A> must be a coset geometry of rank 2. This function computes
the gonality, point and line diameter of <A>cg</A>. 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).
</Description>
</ManSection>

The following example illustrates <F>Rank2Parameters</F>. It uses <Package>AtlasRep</Package> to 
fetch the second Janko group, also known as the Hall-Janko group. Beware that <Package>AtlasRep</Package> needs special write permissions on some systems. The constructed geometry has gonality 6 and both diameters equal to 8. It is known as the <E>Cohen-Tits near octagon</E>.

<Example>
<#Include SYSTEM "../examples/include/diagram_nearoctagon.include"> 
</Example>
</Section>
<Section>
<Heading>Automorphisms, Correlations and Isomorphisms</Heading>
An <E>automorphism</E> of an incidence structure &Gamma; is a permutation of the element set of &Gamma;
such that incidence is preserved and types are fixed (i.e. the type of the image of an element
coincides with the type of that element). One way to compute the (full) automorphism group of &Gamma;
is to compute its incidence graph and then use the available <F>nauty</F> machinery to obtain the group.

<ManSection>
<Oper Name="AutGroupIncidenceStructureWithNauty" Arg="cg"/>
<Returns>permutation group isomorphic to the full automorphism group of <A>cg</A>.</Returns>
<Description> 
The group is computed with <F>nauty</F>, which is part of <Package>GRAPE</Package> but has to be compiled on your system before use. The group returned is a permutation group acting on the set <F>[1..Sum(TypesOfElementsOfIncidenceStructure(cg), t -> NrElementsOfIncidenceStructure(cg,t))]</F>, which is exactly the vertex set of <F>IncidenceGraph(cg)</F>. At the moment the action of the automorphism group on <F>cg</F> is not provided but it can be recovered from the knowledge that the vertex set of <F>IncidenceGraph(cg)</F> first contains all elements of type 1 in <F>cg</F>, then all elements of type 2, etc. or, better still, with the <Package>GRAPE</Package> command <F>VertexNames</F> (see example below).
</Description>
</ManSection>

<ManSection>
<Oper Name="CorGroupIncidenceStructureWithNauty" Arg="cg"/>
<Returns>permutation group isomorphic to the full automorphism group of <A>cg</A>.</Returns>
<Description> 
The group is computed with <F>nauty</F>, which is part of <Package>GRAPE</Package> but has to be compiled on your system before use. The group returned is a permutation group acting on the set <F>[1..Sum(TypesOfElementsOfIncidenceStructure(cg), t -> NrElementsOfIncidenceStructure(cg,t))]</F>, which is exactly the vertex set of <F>IncidenceGraph(cg)</F>. At the moment the action of the automorphism group on <F>cg</F> is not provided but it can be recovered from the knowledge that the vertex set of <F>IncidenceGraph(cg)</F> first contains all elements of type 1 in <F>cg</F>, then all elements of type 2, etc. or with  the <Package>GRAPE</Package> command <F>VertexNames</F>.
</Description>
</ManSection>

<Example>
<#Include SYSTEM "../examples/include/diagram_autcor.include"> 
</Example>

<ManSection>
<Oper Name="IsIsomorphicIncidenceStructureWithNauty" Arg="cg1, cg2"/>
<Returns>true iff cg1 and cg2 are isomorphic.</Returns>
<Description> 
We use <F>nauty</F>, which is part of <Package>GRAPE</Package> but has to be compiled on your system before use. Isomorphism is tested (with <F>nauty</F>) after converting the coset geometries <A>cg1</A> and <A>cg2</A> to coloured graphs.
</Description>
</ManSection>


<Example>
<#Include SYSTEM "../examples/include/diagram_autiso.include"> 
</Example>

</Section>
<Section>
<Heading>Diagrams</Heading>

The <E>diagram</E> 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.<P/>

Technically, the diagram is added to a <F>CosetGeometry</F> object as a mutable attribute. Also the list of rank 2 residues 
of the geometry is added as an attribute once these have been computed. This is done with the operations <F>Rank2Residues</F> 
(to add the attribute) and <F>MakeRank2Residue</F> (to actually compute the resiudes). These operations are not for everyday use and hence remain undocumented.<P/> 

Since the geometry is flag-transitive, all chambers are
equivalent. Let's fix a chamber <Alt Not="HTML"><M>C=\{c_{1}, c_{2},\ldots,c_{n}\}</M></Alt><Alt Only="HTML MathJax"><M>C=\{c_{1}, c_{2},\ldots,c_{n}\}</M></Alt>
<Alt Only="HTML noMathJax">C = &#123; c&lt;sub>1&lt;/sub>,c&lt;sub>2&lt;/sub>,...,c&lt;sub>n&lt;/sub>&#125;</Alt>,
with <Alt Not="HTML"><M>c_{i}</M></Alt><Alt Only="HTML MathJax"><M>c_{i}</M></Alt><Alt Only="HTML noMathJax">c&lt;sub>i&lt;/sub></Alt> of type <M>i</M>. For each subset
<Alt Not="HTML"><M>\{i,j\}</M></Alt><Alt Only="HTML MathJax"><M>\{i,j\}</M></Alt><Alt Only="HTML noMathJax">&#123;i,j&#125;</Alt> of
size two in <Alt Not="HTML"><M>I=\{1,2,\ldots,n\}</M></Alt><Alt Only="HTML MathJax"><M>I=\{1,2,\ldots,n\}</M></Alt><Alt Only="HTML noMathJax">&#123;1,2,...n&#125;</Alt>
we take the residue of the flag <Alt Not="HTML"><M>C\setminus \{c_{i},c_{j}\}</M></Alt><Alt Only="HTML MathJax"><M>C\setminus \{c_{i},c_{j}\}</M></Alt>
<Alt Only="HTML noMathJax">C \ &#123; c&lt;sub>i&lt;/sub>, c&lt;sub>j&lt;/sub>&#125;</Alt>. Flag transitivity ensures
that <E>all</E> residues of type <M>\{i,j\}</M> 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 <M>i</M>, we also define the <M>i</M>-order
to be one less than the number of elements of type <M>i</M> in the residue of a(ny) flag of 
type <M>I\setminus\{i\}</M>. All this information is depicted in
a <E>diagram</E> which is bascically a labelled graph with vertex set <M>I</M>
and edges whenever the point diamater, the line diameter and the
gonality are all greater than <M>2</M>.

<ManSection>
<Oper Name="DiagramOfGeometry" Arg="Gamma"/>
<Returns>the diagram of the geometry <A>Gamma</A></Returns>
<Description> 
<A>Gamma</A> must be a flag-transitive coset geometry.<P/> 

The flag-transitivity is not tested by this operation because such a test can be
time consuming. The command <C>IsFlagTransitiveGeometry</C> can be
used to check flag-transitivity if needed.
</Description>
</ManSection>
<ManSection>
<Oper Name="GeometryOfDiagram" Arg="diag"/>
<Returns>the geometry of which <A>diag</A> is the diagram</Returns>
<Description> 
<A>diag</A> must be a diagram object.
</Description>
</ManSection>

<ManSection>
<Oper Name="DrawDiagram" Arg="diag,filename"/>
<Oper Name="DrawDiagram" Arg="diag,filename,vertexverbosity"/>
<Oper Name="DrawDiagram" Arg="diag,filename,vertexverbosity,edgeverbosity"/>
<Returns>does not return anything but writes a file <A>filename</A>.ps</Returns>
<Description> 
<A>diag</A> must be a diagram. Writes a file <A>filename</A>.ps in the
current directory with a pictorial version of the diagram. This
command uses the <Package>graphviz</Package> package which is
available from http://www.graphviz.org. <P/>

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

By default the diagram provides for each type <M>i</M> the <M>i</M>-order and the number of elements of type <M>i</M>. This behaviour can be changed by providing a <A>vertexverbosity</A> level. A value <A>2</A> results in no label under the vertices and a value <A>1</A> gives only the <M>i</M>-order. Any other potitive integer value yields the default behaviour. 

The default labels for the edges of the diagram use the standard convention that a <M>[g,dp,dl]</M>-gon with all three parameters equal is labelled only with the number <M>g</M>. Putting <A>edgeverbosity</A> equal to <M>2</M> puts no labels at all. This yields the so called ``basic diagram'' of the geometry. With <A>edgeverbosity</A> equal to any integer greater than <M>2</M> all labels contain girth and both diameters.
</Description>
</ManSection>

<Alt Only="LaTeX">\medskip</Alt>

We illustrate the diagram feature with
Neumaier's <Alt Not="HTML"><M>A_8</M></Alt><Alt Only="HTML">A&lt;sub>8&lt;/sub></Alt>-geometry.

The affine space of dimension 3 over the field with two elements is
denoted by <M>AG(3,2)</M>. If we fix a plane <Alt Not="HTML"><M>\Pi</M></Alt><Alt Only="HTML">&#928;</Alt> in <M>PG(3,2)</M>, the
structure induced on the 8 points not in <Alt Not="HTML"><M>\Pi</M></Alt><Alt Only="HTML">&#928;</Alt> by the lines and planes
of <M>PG(3,2)</M> is isomorphic to <M>AG(3,2)</M>. Since every two points of
<M>AG(3,2)</M> define a line, the collinearity graph of <M>AG(3,2)</M> (that is the graph
whose vertices are the points of <M>AG(3,2)</M> and in which two vertices
are adjacent whenever they are collinear) is the complete graph <Alt Not="HTML"><M>K_8</M></Alt><Alt Only="HTML">K&lt;sub>8&lt;/sub></Alt>
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 <Alt Not="HTML"><M>K_8</M></Alt><Alt Only="HTML">K&lt;sub>8&lt;/sub></Alt>. The affine space has 14 planes of 4 points and
there are 70 subsets of 4 elements in the vertex set of <Alt Not="HTML"><M>K_8</M></Alt><Alt Only="HTML">K&lt;sub>8&lt;/sub></Alt>. Each
time we label <Alt Not="HTML"><M>K_8</M></Alt><Alt Only="HTML">K&lt;sub>8&lt;/sub></Alt>, there are 14 of the 70 sets of 4 elements which
become planes of <M>AG(3,2)</M>. The remaining 4-subsets will be called 
<E>nonplanes</E> 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 <M>1</M> is the famous 
<Alt Not="HTML"><M>A_7</M></Alt><Alt Only="HTML">A&lt;sub>7&lt;/sub></Alt>-geometry which is known to be the only flag-transitive locally
classical <M>C_3</M>-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 <Alt Not="HTML"><M>K_8</M></Alt><Alt Only="HTML">K&lt;sub>8&lt;/sub></Alt>,
the elements of type 3 are the 4-subsets of the vertex set 
of <Alt Not="HTML"><M>K_8</M></Alt><Alt Only="HTML">K&lt;sub>8&lt;/sub></Alt> and the elements of type 4 are the 15 nonequivalent labelings
of <Alt Not="HTML"><M>K_8</M></Alt><Alt Only="HTML">K&lt;sub>8&lt;/sub></Alt>. Incidences are mostly the natural ones. A 4-subset
is incident with a labeling of <Alt Not="HTML"><M>K_8</M></Alt><Alt Only="HTML">K&lt;sub>8&lt;/sub></Alt> if it is the set of points of a
nonplane in the model of <M>AG(3,2)</M> defined by the labeling.
<Example>
<#Include SYSTEM "../examples/include/diagram_neumaier.include"> 
</Example>
The produced diagrams are included here: Neumaier's <Alt Not="HTML"><M>A_8</M></Alt><Alt Only="HTML">A&lt;sub>8&lt;/sub></Alt><P/>
<Alt Only="HTML">&lt;img src="./graphics/neuma8.jpg">&lt;/img></Alt><Alt Only="LaTeX">\includegraphics{./graphics/neuma8.pdf}</Alt><P/><P/>
The <Alt Not="HTML"><M>A_7</M></Alt><Alt Only="HTML">A&lt;sub>7&lt;/sub></Alt> geometry:<P/>
<Alt Only="HTML">&lt;img src="./graphics/a7geo.jpg">&lt;/img></Alt><Alt Only="LaTeX">\includegraphics[scale=0.5]{./graphics/a7geo.pdf}</Alt><P/><P/>

On a UNIX system we can start an external viewer (``gv'' or ghostview in this case) from within GAP with the <F>Exec</F> command.

<ManSection>
<Oper Name="DrawDiagramWithNeato" Arg="diag,filename"/>
<Returns>does not return anything but writes a file <A>filename</A>.ps</Returns>
<Description> 
<A>diag</A> must be a diagram. Writes a file <A>filename</A>.ps in the
current directory with a pictorial version of the diagram. <P/>

This command uses a "spring tension" algorithm to draw the diagram <A>diag</A> with straight edges. For some diagrams this looks 
better than the result of <F>DrawDiagram</F>. However this algorithm does not print the vertex labels.<P/>

This
command uses the <Package>graphviz</Package> package which is
available from http://www.graphviz.org. 
In case <Package>graphviz</Package> is not available on your system,
you will get an friendly error message and a file <A>filename</A>.dot
will be written. You can then compile this file later or ask a friend
to help you.
An <M>E_6</M> geometry for comparison: on the left hand side we have the
output of DrawDiagram and on the right hand side we see the result of
DrawDiagramWithNeato<P/>
<Alt Only="HTML">&lt;img src="./graphics/E6.jpg">&lt;/img> &lt;img src="./graphics/E6bis.jpg">&lt;/img></Alt>
<Alt Only="LaTeX">\includegraphics[scale=0.5]{./graphics/E6.jpg} \includegraphics[scale=0.5]{./graphics/E6bis.jpg}</Alt><P/><P/>
</Description>
</ManSection>

</Section>
</Chapter>
