This week John continued his discussion on triality and generalised hexagons from last week. This is also the first week where I have used Luca Trevisan’s LaTex to WordPress program to write up the post.

Recall that we had an eight-dimensional vector space ${V}$ over ${\mathrm{GF}(q)}$ equipped with a quadratic form ${Q(x)=x_0x_4+x_1x_5+x_2x_6-x_3x_7}$, where ${x=(x_0,x_1,\ldots,x_7)\in V}$. We were able to divide the totally isotropic solids into two classes, greeks and latins, so that two solids are in the same class if their intersection is a line or trivial. The points, lines, greeks and latins give rise to the Oriflamme geometry where incidence is inclusion and a greek is incident with a latin if their intersection is a plane.

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.
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