Up to duality, there are two known families of finite (thick) generalised hexagons:

1. the Split Cayley hexagons of order $(q,q)$ related to the Dickson exceptional groups $G_2(q)$.
2. the Twisted Triality hexagons of order $(q,q^3)$ related to the Steinberg exceptional groups $\,^3D_4(q)$.

Since I will only consider generalised hexagons up to duality, I will assume that the order $(s,t)$ of a given one has $s\le t$. The parameter $s$ is one less than the number of points on a line, and $t+1$ is the number of lines on a point. To date, we do not know much about the possible values of these positive integers $s$ and $t$. Here is what we know:

• $s\le t^3$ and $t\le s^3$ (Haemers and Roos, 1981)
• $s^2+st+t^2$ divides $s^3(s^2t^2+st+1)$ (from the multiplicities of the eigenvalues of the point graph).
• There are two (up to duality) generalised hexagons with $s=2$ and they have $t\in\{2,8\}$ respectively (Cohen and Tits 1985).

We do not know that $s+1$ divides $t+1$, even though the known examples satisfy this simple divisibility relation.

Here is a cool number theoretic thing I’ve observed recently, but I can’t (quite yet) prove that it is true:

Claim: If $s,t$ are integers greater than 1, satisfying $s+1 \mid t+1$ and $s^2+st+t^2\mid s^3(s^2t^2+st+1)$, then apart from a small finite number of exceptions, we have $t=s$ or $t=s^3$.

I claim that the only exception is $(s,t)=(14,224)$.

We know that the known examples satisfy the extra relation $s+1\mid t+1$, and it is conceivable that such a simple looking relation holds for generalised hexagons and that it arises for some combinatorial reason. It’s simplicity is the only reason to believe that it might hold, but wouldn’t it be cool if every generalised hexagon satisfied it?

Below is a guest blog post of Melissa Lee, who took part in a six-week summer vacation research project supported by the Australian Mathematical Sciences Institute.

Hi everyone! My name is Melissa Lee and I’ve recently started honours with John and Dr. Eric Swartz here at UWA. This past summer I have been working with John on an AMSI Vacation Research Scholarship. My AMSI VRS project was based on looking at structures embedded in an affine space by viewing them in terms of a game called SET. Continue reading “AMSI Summer Vacation Scholarship”

I’m writing this post so that I can direct my students to it as I often have to go over the evolution of the concepts of ovoids of projective spaces and polar spaces, and to explain that they are (i) different, and (ii) connected. It’s  a bit dry, but it will serve as a reference for future posts.

## Ovoids of projective spaces

I will start with a result of Jacques Tits in 1962, though ovoids arose earlier in the work of Bose and Qvist. An ovoid of a projective space $\mathcal{S}$ is a set of points $\mathcal{O}$ such that for any point $P$ of $\mathcal{O}$, the union of the lines incident with $P$ that are tangent to $\mathcal{O}$ forms a hyperplane of $\mathcal{S}$. Tits showed that an ovoid of $PG(n,q)$ exists if and only if $n\le 3$. For $n\le 3$, we should be careful to stipulate that $q>2$, since the size of an ovoid of $PG(3,2)$ is just 5 and things are a bit messy to state. By a simple counting argument, the number of points of an ovoid of $PG(3,q)$ is $q^2+1$, and for a projective plane of order $q$, an ovoid has size $q+1$. For example, the elliptic quadric of $PG(3,q)$ is an example of an ovoid, and a non-singular conic is an example of an ovoid of $PG(2,q)$. For more on the elliptic quadric example, see this post.

In the planar case, we now call these objects ovals, and reserve the name ‘ovoid’ just for the 3-dimensional case. Now an ovoid and oval have the property that no 3 points are collinear. We have the following way to think of an ovoid in three different ways:

Let $\mathcal{O}$ be a set of points of $PG(3,q)$. The following are equivalent:

1. No 3 points of $\mathcal{O}$ are collinear and $|\mathcal{O}| = q^2+1$;
2. For any point $P$ of $\mathcal{O}$, the union of the lines incident with $P$ that are tangent to $\mathcal{O}$ span a plane of $PG(3,q)$;
3. At every point $P$ of $\mathcal{O}$, there is a unique plane that meets $\mathcal{O}$ only in $P$ (i.e., a tangent plane).

## Ovoids of polar spaces

Jef Thas defined ovoids of polar spaces in his seminal paper “Ovoidal translation planes” in 1972. It is a set of points such that every maximal totally isotropic subspace meets it in precisely one point. Alternatively, we could define it as a set of points no two collinear, with size the number of points divided by the number of points in a maximal. We have:

Let $\mathcal{O}$ be a set of points of a finite polar space $\mathcal{S}$, and let $\mu$ be the number of points of $\mathcal{S}$ divided by the number of points lying in a maximal totally istropic subspace. The following three are equivalent:

1. No 2 points of $\mathcal{O}$ are collinear and $|\mathcal{O}| = \mu$;
2. Every maximal totally isotropic subspace has exactly one point of $\mathcal{O}$ incident with it.
3. At every point $P$ of $\mathcal{O}$, there exists a maximal totally isotropic subspace through $P$ tangent to $\mathcal{O}$.

So where did Thas’ definition come from? The most impressive result of his 1972 paper is the connection between ovoids of the rank 2 symplectic space and ovoids of the 3-dimensional projective spaces.

Theorem (Thas 1972): Let $q$ be an even prime power. Then an ovoid of $W(3,q)$ is also an ovoid of $PG(3,q)$. Conversely, if $\mathcal{O}$ is an ovoid of $PG(3,q)$, then we can define a null polarity $\rho$ defining a $W(3,q)$ such that $\mathcal{O}$ are absolute points for $\rho$.

This result is quite remarkable. It certainly is easier to study and enumerate ovoids of $W(3,q)$, than in $PG(3,q)$. The null polarity hinted at by the theorem is truly beautiful and was first known to Segre (1959): given a point $P$ of $\mathcal{O}$, we define $P^\rho$ to be the unique tangent plane at $P$; for each secant plane $\pi$ of $\mathcal{O}$ we define $\pi^\rho$ to be the nucleus of the oval carved out (i.e., the nucleus of $\mathcal{O}\cap\pi$).

The situation for $q$ odd was already classified independently by Barlotti and Panella (1965): the only ovoids in this case are quadrics. So the open problem on classifying ovoids of $PG(3,q)$ boils down to the even case.

Another reason for defining ovoids this way is that it encapsulates natural examples. A non-degenerate hyperplane section of the Hermitian polar space $H(3,q^2)$ is an ovoid, and a non-degenerate hyperplane section of minus type of the parabolic quadric $Q(4,q)$ is an ovoid. Plus, we can think of the non-singular conic as a rank 1 polar space $Q(2,q)$, and hence an ‘ovoid’ of $Q(2,q)$ coincides trivially with an oval of $PG(2,q)$.

In a future post, I’ll talk about the open problems on ovoids of polar spaces.

The Minister for Education (finally) announced the ARC Future Fellowships commencing in 2013 this morning, and simultaneously, the DECRA’s and Discovery Projects. Our research group missed out on a grant of the first and second kind, but we had success with two Discovery Projects:

1. Cheryl Praeger, Stephen Glasby, and Alice Niemeyer (“Finite linearly representable geometries & symmetries).
2. Gordon (“Real chromatic roots of graphs & matroids”)

I remember last summer when both grant applications were being written, and it was a lot of work for all involved, so it’s great to see that both have awarded to support cutting edge research.

This is a continuation of my last post on this subject. As Gordon remarked in one of his posts, you may need to refresh your browser if some of the embedded gifs do not appear as they should.

### Dualities and isomorphisms of classical groups

Four of the five families of classical generalised quadrangles come in dual pairs: (i) $Q(4,q)$ and $W(3,q)$; (ii) $H(3,q^2)$ and $Q^-(5,q)$. Both can be demonstrated by the Klein correspondence. Recall from the last post that the Klein correspondence maps a line of $\mathsf{PG}(3,q)$ represented as the row space of

$M_{u,v}:=\begin{bmatrix}u_0&u_1&u_2&u_3\\v_0&v_1&v_2&v_3\end{bmatrix}$

to the point $(p_{01}:p_{02}:p_{03}:p_{12}:p_{31}:p_{23})$ where

$p_{ij}=\begin{vmatrix}u_i&u_j\\v_i&v_j\end{vmatrix}$.

Now consider the symplectic generalised quadrangle $W(3,q)$ defined by the bilinear alternating form

$B(x,y):=x_0y_1 - x_1y_0 + x_2y_3 - x_3y_2.$

A totally isotropic line $M_{u,v}$ must then satisfy

$0 = B(u,v) = p_{01} + p_{23}$.

Therefore, the lines of $W(3,q)$ are mapped to points of $Q^+(5,q)$ lying in the hyperplane $\pi:X_0+X_5=0$. Now the quadratic form defining $Q^+(5,q)$ is $Q(x)=x_0x_5+x_1x_4+x_2x_3$ and $\pi$ is the tangent hyperplane at the projective point $(1:0:0:0:0:1)$, which does not lie in the quadric. Hence the hyperplane $\pi$ is non-degenerate and so we see that $W(3,q)$ maps to points of $Q(4,q)$. That this mapping is bijective follows from noting that the number of lines of $W(3,q)$ is equal to the number of points of $Q(4,q)$ (namely, $(q+1)(q^2+1)$).

Hence $\mathsf{P\Gamma Sp}(4,q)\cong \mathsf{P\Gamma O}(5,q)$.

Now we will consider a more difficult situation which reveals that the generalised quadrangles $H(3,q^2)$ and $Q^-(5,q)$ are also formally dual to one another. Continue reading “The Klein Correspondence II”

One of the central and important concepts in projective geometry is the beautiful connection between 3-dimensional projective space and the Klein quadric. As is indicated by the title, this correspondence between these two geometries was named after the German mathematician Felix Klein, who studied it in his dissertation Über die Transformation der allgemeinen Gleichung des zweiten Grades zwischen Linien-Koordinaten auf eine kanonische Form (1868). The Klein Correspondence can be used to give geometric understanding for certain isomorphisms of low rank classical groups, and we will give an example of some of these in a follow-up post. In geometry, the Klein Correspondence can sometimes illuminate an obscure object in projective 3-space, for the Klein quadric is naturally embedded in a 5-dimensional projective space where there is an added richness to the geometry, and one has available other ways to distinguish certain configurations. For example, if you were to learn about linear complexes in 3-dimensional projective space, you might find one class known as the “parabolic congruences” as a somewhat messy object of pencils of lines centered on the points of a common line. However, under the Klein Correspondence, a parabolic congruence becomes a quadratic cone: one of the first 3-dimensional geometric objects we first encountered in school mathematics. Notice that I have not specified the field we are working over; it won’t matter!

The day started badly. I woke at 3am to a sick and miserable one-year-old, I got stuck in a traffic jam on the way to work, and by lunchtime I was feeling down about the usual whackacademic innuendo. But at 3pm, Eric Swartz knocked on my door. Eric has stunningly proved an outstanding conjecture on generalised quadrangles which I’ll report on when I have his blessing.

Rather than let this blog die off due to lack of posts, I thought I’d write about a small but annoying problem that I’ve been thinking about recently, but without success, and a natural conjecture that has arisen from it.

[Note: On some browsers there appears to be a sizing problem with the images WordPress uses for LaTeX formulas, and they appear ten times the right size. If this happens, reloading the webpage seems to fix it]

The problem arose in the context of binary matroids, but because a binary matroid is really just a set of points in a binary vector space, it can be phrased entirely as a linear algebra problem.

So let ${M}$ be a set of non-zero vectors in the vector space ${V = GF(2)^r}$ such that ${M}$ spans ${V}$ and, for reasons to be clarified later, no vector in ${M}$ is independent of the others. In matroid terminology, this just says that ${M}$ is a simple binary matroid of rank ${r}$ with no coloops.

Then define a  basis of ${M}$ to be a linearly independent subset of ${M}$ of rank ${r}$ (in other words, just a basis of ${V}$ and a circuit of ${M}$ to be a minimally dependent set of vectors, i.e. a set of vectors that is linearly dependent but any proper subset of which is linearly independent.

As an example, take ${M = PG(2,2)}$ (a.k.a the Fano plane) — this means to take all the non-zero vectors in ${GF(2)^3}$. This has 28 bases (7 choices for a first vector ${v}$, 6 for a second vector ${w}$ and then 4 for the third vector which cannot be ${v}$, ${w}$ or ${v+w}$ , then all divided by 6 because this counts each basis ${6}$ times). It has 7 circuits of size 3 (each being of the form ${v}$, ${w}$, ${v+w}$) and 7 circuits of size 4, being the complements of the circuits of size 3, for a total of 14 circuits.

Letting ${b(M)}$, ${c(M)}$ denote the numbers of bases and circuits of ${M}$ respectively, the question is about the ratio of these two numbers. More precisely,

Determine a lower bound, in terms of the rank ${r}$, for the ratio ${\frac{b(M)}{c(M)}}$?

Over the summer break, Gordon and I each supervised a 2nd year undergraduate student in a research project. At UWA, the first course in group theory is in 3rd year, and we do not teach any combinatorics, so we needed to give each student a crash course before they could sink their teeth into a research problem. One of their outcomes was a blog post, and the first of these is by my student, who was also supervised by Sylvia Morris. His project was on “Sets of type $(m,n)$ in projective spaces”.