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.

A correlation of a geometry is a permutation of its elements which preserves incidence but unlike a collineation, we allow it to permute the types.  A duality is a correlation which induces a permutation of order two on the nodes of the diagram of the geometry, while a polarity is a duality which is a permutation of order 2 of the elements of the geometry. We discussed dualities and polarities of projective spaces in an earlier session.

Jacques Tits defined a T-collinéation of a geometry to be a collineation which induces a permutation of order three on the nodes of the diagram of the geometry. A triality is then a T-collinéation  which is a permutation of order three on the elements. This is a bit unfortunate as it doesn’t correspond to the duality/polarity situation. Moreover, many authors (including myself) use the term triality to mean a T-collinéation.

Let $V =GF(q)^8$ and let Q be the quadratic form on V defined by $Q(v)=x_0x_4+x_1x_5+x_2x_6-x_3x_7$ where $v=(x_0,x_2,\ldots,x_7)$. We can then create the geometry given by

• points : the totally singular 1-spaces
• lines: the totally singular 2-spaces
• planes: the totally singular 3-spaces
• solids: the totally singular 4-spaces

and incidence is the usual inclusion. This gives the classical polar space often denoted by $Q^+(7,q)$ and with collineation group $P\Gamma O^+(8,q)$. It has diagram $B_4$.

This configuration can also be used to define the Oriflamme geometry: We can define an equivalence relation on the set of solids such that $U\sim W$ if and only is $U\cap W$ has even codimension in $U$, that is, $U\cap W=\{0\}$ or is a line. This relation has two equivalence classes referred to as greeks and latins. The two equivalence classes are in fact the two orbits of $P\Omega^+(8,q)$ on solids. (The group $P\Omega^+(8,q)$ is the derived subgroup of $PSO^+(8,q)$.) We say that a greek is incident with a latin if they intersect in a plane. The rest of the incidence is that inherited from the polar space. This gives us a rank 4 geometry with diagram

Now this geometry has a triality $\theta$ which maps points to greeks, greeks to latins, latins to points and fixes the set of lines, that is, it has order three and cyclically permutes the outer three nodes of the diagram. The absolute subspaces under $\theta$ are those subspaces x such that x is incident with $x^{\theta}$.  The absolute points are those which have $x_3=x_7$ and the absolute lines are all contained in this hyperplane (but are not all of the lines). The absolute points and lines form a generalised hexagon known as the split Cayley hexagon of order (q,q) which has associated group $G_2(q)$.

John didn’t really define what $\theta$ actually is but next week he will hopefully be telling us what the lines of the generalised hexagon actually are. Moroever, the triality $\theta$ which gives the split Cayley hexagon is not the unique triality of our geometry.  There is another choice of triality whose absolute points and lines forms the generalised hexagon of order $(q^3,q)$ associated with the group ${}^3D_4(q)$.

## 4 thoughts on “Buildings, geometries and algebraic groups study group VIII”

1. Alice Devillers says:

I think the answer to your question is that it does not imply bipartite. But I had stronger hypotheses, like n-arc transitivity.

1. Alice Devillers says:

Higman-Sims had diameter 2 and girth 4 I think. Not bipartite.

2. Yes, I just checked and the Higman-Sims graph has girth 4. I wonder which other strongly regular graphs have girth 4…

3. Just to clarify, it’s not completely true what you wrote about the known finite generalised hexagons and octagons; there are the duals as well. Here are the known examples:

Generalised hexagons:
– split Cayley hexagons of order $(q,q)$ (which are self dual only for q a power of 3)
– dual split Cayley hexagons of order $(q,q)$
– twisted triality hexagons of order $(q^3,q)$
– dual twisted triality hexagons of order $(q,q^3)$

Generalised octagons:
– Ree-Tits octagons of order $(q,q^2)$
– dual Ree-Tits octagons of order $(q^2,q)$