In this post I wish to discuss flock generalised quadrangles. As mentioned in the first of this series, John has already discussed these a bit in a previous post so my main aim will be to flesh that out and provide more background. I have relied heavily on Maska Law’s PhD thesis which is available from Ghent’s PhD theses in finite geometry page.

Recall that a conic ${\mathcal{C}}$ is the set of zeros of a nondegenerate quadratic form on ${\mathrm{PG}(2,q)}$. Embed ${\mathrm{PG}(2,q)}$ as a hyperplane ${\pi}$ in ${\mathrm{PG}(3,q)}$ and take a point ${P}$ not on ${\pi}$. For each of the ${q+1}$ points of ${\mathcal{C}}$ there is a unique line through such a point and ${P}$. Let ${\mathcal{K}}$ be the set of all ${q^2+q+1}$ points on these ${q+1}$ lines. The set ${\mathcal{K}}$ is called a quadratic cone with vertex ${P}$. Now ${\mathsf{P}\Gamma\mathsf{L}(4,q)}$ acts transitively on the set of pairs ${(P,\pi)}$ of points ${P}$ and hyperplanes ${\pi}$ of ${\mathrm{PG}(3,q)}$ where ${\pi}$ does not contain ${P}$, and the stabiliser of such a pair induces ${\mathrm{GL}(3,q)}$ on ${\pi}$ and so acts transitively on the set of conics contained in ${\pi}$ . Thus all quadratic cones of ${\mathrm{PG}(3,q)}$ are equivalent.

An easy way to construct a quadratic cone is to take the zeros of the degenerate quadratic form ${Q(x)=x_2^2-x_1x_3}$, where ${x=(x_1,x_2,x_3,x_4)}$. Here we take ${P}$ to be ${\langle (0,0,0,1)\rangle}$ and ${\pi=\langle (1,0,0,0),(0,1,0,0),(0,0,1,0)\rangle}$. The zeros of ${Q}$ on ${\pi}$ form the conic ${\{\langle (1,t,t^2,0)\rangle\mid t\in\mathsf{GF}(q)\}\cup \{\langle (0,0,1,0) \rangle\}}$. Note that for any plane ${\pi'}$ of ${\mathsf{PG}(3,q)}$ the set of zeros of ${Q}$ on ${\pi'}$ forms a conic. This is all reminiscent of the classical case of a cone in ${\mathbb{R}^3}$, where the intersections of a plane with the cone are the conic sections and are either a point, a circle, an ellipse, a parabola or a hyperbola.

A flock of a quadratic cone ${\mathcal{K}}$ with vertex ${P}$ is a partition of ${\mathcal{K}\backslash\{P\}}$ into ${q}$ disjoint conics. Each conic is the intersection of ${\mathcal{K}}$ with a plane. Let ${\ell}$ be a line of ${\mathrm{PG}(3,q)}$ which intersects ${\mathcal{K}}$ trivially. Then ${\ell}$ is contained in ${q+1}$ planes, one of which contains ${P}$. Each of the remaining ${q}$ planes containing ${\ell}$ meets each of the lines which make up ${\mathcal{K}}$ and hence meets ${\mathcal{K}}$ in ${q+1}$ points. Such a set of ${q+1}$ points is a conic. Morever, since the intersection of two planes through ${\ell}$ is ${\ell}$, the ${q}$ conics we obtain are all disjoint and so we get a flock. Such a flock is called a linear flock.

BLT-sets

A BLT-set of lines of ${W(3,q)}$ is a set ${\mathcal{L}}$ of ${q+1}$ disjoint lines such that no line of ${\mathsf{W}(3,q)}$ meets more than two lines of ${\mathcal{L}}$. BLT-sets are named after Laura Bader, Guglielmo Lunardon and Jef Thas who first studied them in 1990. They only exist for ${q}$ odd.

Now let ${\ell}$ be a line of ${\mathsf{W}(3,q)}$ which meets some line ${\ell_1}$ of ${\mathcal{L}}$. Let ${P}$ be the point of intersection and suppose that ${\ell}$ does not meet any of the remaining ${q}$ lines of ${\mathcal{L}}$. Then for each line ${\ell_2}$ of ${\mathcal{L}\backslash\{\ell_1\}}$, the GQ property implies that for each point ${Q}$ on ${\ell_2}$, ${Q}$ is collinear with a unique point of ${\ell}$ and in particular there is a unique point ${Q'}$ of ${\ell_2}$ collinear with ${P}$. Considering each of the lines of ${\mathcal{L}\backslash\{\ell_1\}}$ we obtain ${q}$ points collinear with ${P}$. Since no line of ${\mathsf{W}(3,q)}$ meets more than 2 lines of ${\mathcal{L}}$, it follows that this gives ${q}$ lines incident with ${P}$. These ${q}$ lines are distinct from ${\ell_1}$ and ${\ell}$. Thus we have obtained ${q+2}$ lines incident with ${P}$, contradicting the fact that each point of ${\mathsf{W}(3,q)}$ lies on exactly ${q+1}$ lines. Hence ${\ell}$ meets two lines of ${\mathcal{L}}$, that is, every line of ${\mathsf{W}(3,q)}$ meets 0 or 2 lines of ${\mathcal{L}}$.

An example of a BLT-set is the linear one and is constructed as follows. Let ${V}$ be a 2-dimensional vector space over ${\mathsf{GF}(q^2)}$ equipped with a nondegenerate hermitian form ${B}$. Then ${V}$ is also a 4-dimensional vector space over ${\mathsf{GF}(q)}$and we can define a new form ${\overline{B}:V\times V\rightarrow \mathsf{GF}(q)}$ such that ${\overline{B}=Tr\circ B}$, where ${Tr:\mathsf{GF}(q^2)\rightarrow\mathsf{GF}(q)}$ is the trace map which maps each field element ${\lambda}$ to ${\lambda+\lambda^q}$. Then ${\overline{B}}$ is a nondegenerate alternating form. Moreover, the ${q+1}$ one-dimensional subspaces over ${\mathsf{GF}(q^2)}$ which are totally singular with respect to ${B}$ are totally singular two-dimensional subspaces over ${\mathsf{GF}(q)}$ with respect to ${\overline{B}}$.

Since ${\mathsf{W}(3,q)}$ is dual to ${\mathsf{Q}(4,q)}$, we could equivalently study BLT-sets of points of ${\mathsf{Q}(4,q)}$ where they are sets of ${q+1}$ points such that each point of ${\mathsf{Q}(4,q)}$ is collinear with at most 2 points in the set. This is the setting in which BLT-sets were originally studied.

${q}$-clans

A matrix ${A}$ is called anisotropic if the only vector ${x}$ such that ${xAx^T=0}$ is the zero vector. A ${q}$-clan is a set of ${q}$ ${2\times 2}$ matrices over ${\mathsf{GF}(q)}$ such that the difference of any two distinct matrices is anisotropic.

The classical ${q}$-clan is

$\displaystyle \left\{\begin{pmatrix} t&0\\ 0&-nt\end{pmatrix} \Big\vert t\in\mathsf{GF}(q)\right\}$

for ${n}$ is a nonsquare in ${\mathsf{GF}(q)}$.

Constructing GQs

In 1980, Payne showed that a ${q}$-clan gives rise to a generalised quadrangle of order ${(q^2,q)}$ via a group coset construction due to Kantor. (This will be a part of my next post.) In 1987, Thas then showed that flocks of quadratic cones give rise to ${q}$-clans and vice versa.

The original motivation of studying BLT-sets by Bader, Lunardon and Thas in 1990 was that for ${q}$ odd, a BLT-set gives rise to ${q+1}$ flocks of the quadratic cone, while conversely a flock of the quadratic cone gives rise to a BLT-set. The number of equivance classes of flocks arising from a given BLT-set is the number of orbits of the stabiliser of the BLT-set on the lines of the BLT-set.

So the three seemingly different objects of flocks, ${q}$-clans and BLT-sets are all intimately connected. This connection links the classical ${q}$-clan with the linear flock and linear BLT-set. The generalised quadrangles associated with these three objects are called flock quadrangles.

When ${q}$ is odd the flock quadrangle can be constructed directly from the BLT-set via a construction of Knarr published in 1992. Despite the fact that a BLT-set gives rise to ${q+1}$ flocks and these flocks may not be equivalent, it only gives rise to one generalised quadrangle. In 2001, Thas came up with a geometrical constuction of the flock GQ’s from the flock for all ${q}$. This construction gives the same GQ as the Knarr construction when ${q}$ is odd.

John outlined the Knarr construction in a previous post. He actually constructed it in a way that is nonstandard in the literature (but proved useful in our work showing that every flock quadrangle contains a hemisystem). In this post I will use the standard construction.

We start in the symplectic polar space ${\mathsf{W}(5,q)}$ of rank ${3}$ which arises from taking the totally isotropic one, two and three-dimensional vector subspaces of ${\mathsf{GF}(q)^6}$ with respect to an alternating bilinear form. For convenience we will use the form defined by

$\displaystyle \beta(x , y ) = x_1y_6-x_6y_1+x_2y_5-x_5y_2+x_3y_4-x_4y_3.$

In particular ${\beta(x,y)=xJ'y^T}$ where

$\displaystyle J'=\begin{pmatrix} 0&0&0&0&0&1\\ 0&0&0&0&1&0\\ 0&0&0&1&0&0\\ 0&0&-1&0&0&0\\ 0&-1&0&0&0&0\\ -1&0&0&0&0&0 \end{pmatrix}$

For an arbitray subspace ${U}$ of the ambient projective space we can define ${U^\perp= \{v\in \mathsf{GF}(q)^6: \beta( u,v ) =0\text{ for all } u\in U\}}$.

Let ${P}$ be a point of ${\mathsf{PG}(5,q)}$. The totally isotropic lines and planes incident with ${P}$ yield the quotient polar space ${P^\perp/P}$ isomorphic to ${\mathsf{W}(3, q)}$. Thus given a BLT-set ${\mathcal{O}}$ of ${\mathsf{W}(3,q)}$ we can identify ${\mathcal{O}}$ with a set of totally isotropic planes on ${P}$. We can then construct a generalised quadrangle ${\mathcal{K}(\mathcal{O})}$ of order ${(q^2,q)}$ as follows. It is another one of these geometrical constructions where there are several sorts of points and lines.

 Points Lines (i) points of ${\mathsf{PG}(5,q)}$ not in ${P^\perp}$ (a) totally isotropic planes not contained (ii) lines not incident with ${P}$ but in ${P^\perp}$ and meeting some element of contained in some element of ${\mathcal{O}}$ ${\mathcal{O}}$ in a line (iii) the point ${P}$ (b) elements of ${\mathcal{O}}$

Incidence is inherited from that of ${\mathsf{PG}(5,q)}$.

There is a more general construction by Shult and Thas, which takes as input an analogue of a BLT-set and constructs a generalised quadrangle. For example, one can define a BLT-set of ${q^2+1}$ lines of ${\mathsf{Q}^-(5,q)}$ with ${q}$ odd and the construction yields a GQ of order ${(q^2,q^2)}$ from such a set. However, the only known BLT-set in ${\mathsf{Q}^-(5,q)}$ is and the GQ constructed in this case is ${\mathsf{Q}(4,q^2)}$.

Examples

So far the only example we have seen of a BLT-set is the linear one. Using this as an input into the Knarr construction we obtain the classical generalised quadrangle ${\mathsf{H}(3,q^2)}$. Indeed ${\mathcal{K}(\mathcal{O})}$ is isomorphic to ${\mathsf{H}(3,q^2)}$ if and only if ${\mathcal{O}}$ is a linear BLT-set. In fact, ${\mathcal{K}(\mathcal{O}_1)}$ is isomorphic to ${\mathcal{K}(\mathcal{O}_2)}$ if and only if the BLT-sets ${\mathcal{O}_1}$ and ${\mathcal{O}_2}$ are equivalent, that is, the first can be mapped to the second by an automorphism of ${\mathsf{W}(3,q)}$.

There are several infinite families of nonlinear BLT-sets. Since they are linked with flocks and q-clans, often a family is attributed to several authors as the associated objects were originally discovered separately. There are also many examples which are believed to be sporadic. A large number were discovered by Maska Law as part of the work for his thesis. Anton Betten has a webpage which classifies the BLT-sets in ${\mathsf{W}(3,q)}$ for ${q\leq 67}$.

There are also several families of nonclassical ${q}$-clans in the ${q}$ even case as well. These are associated with ovals of ${\mathsf{PG}(2,q)}$.

Groups

The post wouldn’t be complete without a discussion of the automorphism groups of the flock generalised quadrangles. Again, I will concentrate on the ${q}$ odd case so that we can use Knarr’s construction.

Let ${G}$ be the semisimilarity group of the form ${\beta}$ given above and let

$\displaystyle H=\{A\in\mathsf{GL}(6,q)\mid AJ'A^T=\lambda J' \text{ for some }\lambda\in\mathsf{GF}(q)\}\cong \mathsf{GSp}(6,q)\}$

the group of similarities. Let

$\displaystyle J=\begin{pmatrix} 0&0&0&1\\ 0&0&1&0\\ 0&-1&0&0\\ -1&0&0&0\\ \end{pmatrix}$

and take ${P}$ to be the span of ${[1,0,0,0,0,0]}$. Then ${H_P=E\rtimes (Q\times R)}$ where

$\displaystyle \begin{array}{rl} E &=\left\{\begin{pmatrix} 1&0&0\\ J^T\textbf{a}^T&I&0\\ z&\textbf{a}&1 \end{pmatrix}\Big\vert \textbf{a}\in\mathsf{GF}(q)^4,z\in\mathsf{GF}(q)\right\}\\ Q &=\left\{\begin{pmatrix} \lambda&0&0\\ 0&I&0\\ 0&0&\lambda^{-1} \end{pmatrix}\Big\vert \lambda\in\mathsf{GF}(q)\backslash\{0\}\right\}\cong C_{q-1}\\ R &=\left\{\begin{pmatrix} \lambda&0&0\\ 0&A&0\\ 0&0&1 \end{pmatrix}\Big\vert A\in\mathsf{GL}(4,q), AJA^T=\lambda J\right\}\cong\mathsf{GSp}(4,q) \end{array}$

and ${(H_P)_{\mathcal{O}}=E\rtimes (Q\times R_{\mathcal{O}})\cong E\rtimes (Q\times \mathsf{GSp}(4,q)_{\mathcal{O}})}$. Moreover, ${G_P=\langle H_P,\sigma\rangle}$, where ${\sigma}$ is the standard Frobenius automorphism. Note that ${\langle R,\sigma\rangle \cong\Gamma\mathsf{Sp}(4,q)}$ and acts on ${E/Z(E)}$ as in its natural action on a 4-dimensional vector-space over ${\mathsf{GF}(q)}$. Moreover, ${(G_P)_{\mathcal{O}}=E\rtimes (Q\rtimes \langle R,\sigma\rangle_{\mathcal{O}})\cong E\rtimes (Q\rtimes \Gamma\mathsf{Sp}(4,q)_{\mathcal{O}})}$. The group ${(G_P)_{\mathcal{O}}}$ preserves the flock generalised quadrangle ${\mathcal{K}(\mathcal{O})}$ and contains the subgroup ${Z}$ of all scalar matrices. Hence ${E\rtimes \Gamma\mathsf{Sp}(4,q)_{\mathcal{O}}\cong (G_P)_{\mathcal{O}}/Z\leqslant\mathsf{Aut}(\mathcal{K}(\mathcal{O}))}$.

It appears that apart from the classical case, this is the full automorphism group of the ${\mathcal{K}(\mathcal{O})}$. However, I have yet to fully trace this through the literature and John believes that in the case of the Kantor-Knuth flock quadrangles the full automorphism group can be two or four times larger. I am a bit confused here as in these cases it is possible for the stabiliser in ${\Gamma\mathsf{Sp}(4,q)}$ of the associated BLT-set can have a nontrivial kernel in its action on the ${q+1}$ lines of the BLT-set. I will look to clear this up this week. What will be required though is the Fundamental Theorem of ${q}$-clan geometry of Stan Payne.

Added 22/9/10: The full automorphism group is indeed always $E\rtimes \Gamma\mathsf{Sp}(4,q)_{\mathcal{O}}$ in the nonclassical case. The reference is IV.1 and IV.2 in Payne and Thas’s paper `Generalized quadrangles, BLT-sets, and Fisher flocks’, 85 (1991) Congr. Numer. 161-192. Thanks to Tim Penttila for the reference.

Next

The aim for my next post is to discuss elation generalised quadrangles and translation generalised quadrangles. This will include the group coset construction of elation generalised quadrangles and Kantor families. After that I will have outlined all the known GQs.