Buildings, geometries and algebraic groups study group IX

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.

Continue reading “Buildings, geometries and algebraic groups study group IX”

Buildings, geometries and algebraic groups study group VIII

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.

Continue reading “Buildings, geometries and algebraic groups study group VIII”

Up ↑