After our summer break our regular study group resumed on Friday. This week, John started what looks like being a two week discussion about $G_2$ and generalised hexagons.

A generalised polygon (n-gon) of order (s,t) is a 1-(v,s+1,t+1) design (that is, a set of v points and a collection of  (s+1)-subsets called blocks, or lines,  such that each point lies in t+1 blocks) whose incidence graph has girth 2n and diameter n.  A polygon with n vertices and the edges taken to be the blocks, is a generalised n-gon with s=t=1. The incidence graph of such a polygon is a cycle of length 2n. We call a generalised polygon thick if $s,t >1$.

Some authors define a generalised n-gon to be a bipartite graph with girth 2n and diameter n. John does not like this definition as it is referring to the incidence graph and not the design/geometry. This reminds me of a question Alice, Gordon and I had a couple of weeks ago which we were not able to resolve. We were wondering if “bipartite” was necessary in this definition. That is, is a graph of girth 2n and diameter n bipartite? Any answers are more than welcome.

The incidence graph of a generalised 2-gon is a complete bipartite graph. The generalised 3-gons are the projective planes and the generalised 4-gons, or generalised quadrangles, are the rank two polar spaces.

Walter Feit and Graham Higman proved in 1964 (in the first volume of the Journal of Algebra) that finite thick generalised n-gons only exist for $n=2,3,4,6$ and $8$. While there are many projective planes and generalised quadrangles, there are only two known infinite families of generalised hexagons and one known infinite family of generalised octagons.

John led the discussion again this week and looked at coset geometries and polar spaces.

John began with the following pictureThis is an incidence geometry with 15 points and 15 lines each containing 3 points. It is the smallest thick generalised quadrangle and denoted by $W(3,2)$. It has automorphism group $S_6$. This can be seen by taking the points to be the edges of the complete graph $K_6$ and the lines to be the matchings of $K_6$ (that is the sets of three disjoint edges). The incidence graph of this geometry is known as Tutte’s 8-cage or the Tutte-Coxeter graph.

John next drew the Fano planeThis projective plane is denoted by $PG(3,2)$ and is the smallest thick generalised triangle. It has automorphism group $PGL(3,2)\cong PSL(2,7)$. The incidence graph of this geometry is the Heawood graph.

Next we had the smallest (thick) generalised digon, which is the geometry with three points and three lines such that each line consists of all three points. The incidence graph for this geometry is the complete bipartite graph $K_{3,3}$.