John led the discussion last week.  I am a bit late in posting as I have been busy organising the WA Junior Olympiad which was on Saturday.

The topic this week is what is called Buekenhout geometries or diagram geometries. We also have a new name.

As we have seen so far, a projective space can be associated with the Dynkin diagram of type $A_n$ and polar spaces can be associated with the Dynkin diagram of type $B_n$. Tits’ theory of buildings generalises these notions and associates to each simple group of Lie type a geometry known as a building and this building can be encoded in the Dynkin diagram associated with the group. In his seminal 1979 paper, `Diagrams for geometries and groups’ in JCTA, Francis Buekenhout looked to generalise this further and introduced the notion of diagram geometries. These are also referred to as Buekenhout geometries.

Let X be a set whose elements we refer to as elements. The set X comes equipped with an incidence relation I which is reflexive and symmetric,  a set $\Delta$ whose elements are called types, and  a map $t:X\rightarrow \Delta$ which is onto.  A flag is  a subset of pairwise incident elements of X. We call the triple $\Gamma= (X,I,t)$ a geometry if it satisfies the following two axioms:

1. Two elements of the same type are incident if and only if they are equal. That is, $xIy$ and $t(x)=t(y)$ if and only if $x=y$
2. Every flag is contained in a maximal flag and a maximal flag contains an element of every type.

The rank of $\Gamma$ is $|\Delta|$, while the rank of a flag F is $|F|$, that is, $|t(F)|$.

Triples which satisfy the first condition and not the second are often called pregeometries.

Given a geometry, we can form the incidence graph whose vertices are the elements of the geometry and two distinct elements are adjacent if and only if they are incident in the geometry. If $|\Delta|=k$, this gives us a k-multipartite graph such that the maximal cliques have size k.

For example, we can take X to be the set of all vertices, edges and faces of a cube, $\Delta=\{\text{ face, edge, vertex }\}$ and $t:X\rightarrow \Delta$ to be the function which describes what each element of X is. This gives us a geometry $(X,I,t)$ of rank 3. If instead we had added a handle to our cube, that is, an extra edge and vertex coming off a corner then we would have a pregeometry which is not a geometry. Given a flag F of a geometry $\Gamma$, the residue $\Gamma_F$ is the geometry whose set of elements is $X_F=\{x\in X\backslash F\mid xIy \text{ for all } y \in F\}$ and which has incidence and type function inherited from $\Gamma$.

We use diagrams to represent rank 2 geometries. We will generally refer to the two types of elements as points and lines and say that a point is on a line if it is incident with it. We describe the common rank 2 geometries below.

A partial linear space is a rank two geometry such that every pair of points lies on at most one line. We represent such geometries by: A linear space is a rank two geometry such that every pair of points lies on a unique line. The following are the two diagrams used to denote such a geometry. The first one has points on the left and lines on the right, while the second has points on the right. The circle geometry, or complete graph geometry, is the geometry whose points are the vertices of a complete graph and whose lines are the edges. This geometry is represented by one of the diagrams. Again, the first one has points on the left and lines on the right, while the second has points on the right. The incidence graph of such a geometry is the graph formed by placing an extra vertex at the midpoint of every edge of the complete graph.

A generalised n-gon is a geometry whose incidence graph has diameter n and girth 2n. If we think of the usual n-cycle, then this gives a geometry whose points are the vertices and lines are the edges. The incidence graph of this geometry is then a 2n-cycle and hence we get a generalised n-gon. It is a theorem of Feit and Higman that if a generalised n-gon is thick, that is, each line contains at least 3 points and each point lies on at least 3 lines then n is one of 2,3,4,6 or 8. A generalised 2-gon (or generalised digon) is just a complete bipartite graph while a generalised 3-gon is a projective plane. We represent a generalised n-gon by the diagram Often we denote a generalised digon by two disjoint nodes , a projective plane by two nodes with a single line between them, a generalised quadrangle by and a generalised hexagon by We denote an affine plane by the diagram. We can associate a diagram to a geometry of higher rank as follows. We form a diagram with nodes the elements of $\Delta$.  Given two types $i,j\in \Delta$ we take a flag F of type $\Delta\backslash\{ i,j\}$ and form the residue $\Gamma_F$. This is a geometry of rank 2 and we connect the nodes i and j according to the class of this geometry. For this to be well defined we require that all rank 2 residues of type $\{i,j\}$ come from the same class. This can be guaranteed by assuming that our geometry is flag-transitive, that is, given any two flags of the same type there is an automorphism (a permutation of X which preserves incidence) which maps one to the other.

If we take our original example of the cube, the rank 2 residues are the residues of a vertex, edge or face as seen below.   Combining these and ordering the types as faces, edges, vertices, the diagram for the cube geometry is An affine space then has the diagram while a projective space has the diagram In fact a geometry with the same diagram as a projective space must be a generalised projective space. A generalised projective space is a lattice which is atomic (every element is a join of atoms) and modular (that is if $x\leq z$ then $x\vee (y\wedge z)=(x\vee y)\wedge z)$ for all elements y.)

We saw in an earlier post that the set of totally isotropic subspaces of a vector space equipped with an alternating form forms a polar space. Other polar spaces can be formed by taking the totally isotropic subspaces of a hermitian form or the totally singular subspaces of a quadratic form. Such polar spaces have  diagram As mentioned in a previous week, there is the Neumaier geometry with diagram the $C_3$ diagram. Aschbacher proved that the only geometries whose automorphism group is flag-transitive, all residue stabilisers are finite  and which have a $C_3$ diagram are polar spaces of rank 3 and the Neumaier geometry.