I am finally onto my last post on generalised quadrangles. The topic of this one is translation generalised quadrangles. These are a special case of elation generalised quadrangles outlined in my last post. The main source for this post has been Michel Lavrauw’s Phd thesis which is availiable from Ghent’s PhD theses in finite geometry page.

Recall that an elation generalised quadrangle (EGQ) is a generalised quadrangle with a base point ${x}$ and a group of automorphisms that fixes each line incident with ${x}$ and acts regularly on the set of points not collinear with ${x}$. A translation generalised quadrangle (or TGQ) is an EGQ with an abelian elation group. In this case the elation group is referred to as a translation group. We saw in the EGQ post, that ${\mathsf{W}(3,q)}$, for ${q}$ even, has an abelian elation group and so is an example of a TGQ. It has been proved by Stan Payne and Jef Thas that for a TGQ, the elation group must be elementary abelian and in particular ${s}$ and ${t}$ are powers of the same prime.

Eggs

An egg of ${\mathrm{PG}(2n+m-1,q)}$ is a set ${\mathcal{E}}$ of ${q^m+1}$ subspaces of projective dimension ${n-1}$ such that any three span a subspace of dimension ${3n-1}$ and each ${E\in\mathcal{E}}$ is contained in a subspace ${T_E}$ of dimension ${n+m-1}$ whose intersection with ${\mathcal{E}}$ is ${E}$. The subspace ${T_E}$ is called a tangent space.

If ${n=m}$ we call an egg a pseudo-oval. When ${n=m=1}$ we have a collection of ${q+1}$ points of ${\mathrm{PG}(2,q)}$ such that no 3 are collinear, that is, we have an oval. The only known examples of pseudo-ovals are constructed by taking an oval of ${\mathrm{PG}(2,q^n)}$ and embedding it in ${\mathrm{PG}(3n-1,q)}$ , by interpreting the underlying 3-dimensional vector space over ${\mathrm{GF}(q^n)}$ as a ${3n}$-dimensional vector space over ${\mathrm{GF}(q)}$. Points of ${\mathrm{PG}(2,q^n)}$ then become ${(n-1)}$-subspaces of ${\mathrm{PG}(3n-1,q)}$, and since any three points of the original oval space a plane of ${\mathrm{PG}(2,q^n)}$they span a subspace of dimension ${3n-1}$ over ${\mathrm{GF}(q)}$. The tangent lines of the oval are then tangent spaces of the pseudo-oval.

When ${2n=m}$ an egg is called a pseudo-ovoid. When ${n=1}$ and ${m=2}$ this gives a collection of ${q^2+1}$ points of ${\mathrm{PG}(3,q)}$ such that no three are collinear, that is, we have an ovoid. Again an ovoid of ${\mathrm{PG}(3,q^n)}$ yields a pseudo-ovoid of ${\mathrm{PG}(4n-1,q)}$. However, there are examples of pseudo-ovoids which do not arise in this manner. There is the Kantor pseudo-ovoid for ${q}$ odd, the Cohen-Ganley pseudo-ovoid for ${q=3}$ and arbitray ${n}$, and the Penttila-Williams pseudo-ovoid for ${q=3}$ and ${n=5}$. When ${n\neq m}$, the ${q^m+1}$ tangent spaces of an egg ${\mathcal{E}}$ forms an egg in the dual space of ${\mathrm{PG}(2n+m-1,q)}$, known as the dual egg and denoted by ${\mathcal{E}^D}$. All known eggs are either pseudo-ovals or pseudo-ovoids. Such eggs are called classical.

Constructing a TGQ from an egg

Recall from an earlier post in this series that given an oval ${\mathcal{O}}$ of ${\mathrm{PG}(2,q)}$ we can construct the generalised quadrangle ${T_2(\mathcal{O})}$. Similarly given an ovoid ${\mathcal{O}}$ of ${\mathrm{PG}(3,q)}$ we can construct the generalised quadrangle ${T_3(\mathcal{O})}$. These two constructions can be generalised so that we can construct a generalised quadrangle from an egg.

Let ${\mathcal{E}}$ be an egg in ${H=\mathrm{PG}(2n+m-1,q)}$. Embed ${H}$ as a hyperplane in ${A=\mathrm{PG}(2n+m,q)}$. We construct a new incidence structure whose points are

• (i) the points of ${A\backslash H}$.
• (ii) the ${(n+m)}$-dimensional subspaces of ${A}$ that meet ${H}$ in a tangent space of ${\mathcal{E}}$.
• (iii) an extra point ${(\infty)}$

and whose lines are

• (a) the ${n}$-dimensional spaces of ${A}$ that meet ${H}$ in an element of the egg ${\mathcal{E}}$.
• (b) the elements of ${\mathcal{E}}$.

Incidence is as follows: a point of type (i) is incident with the lines of type (a) that contain it, a point of type (ii) is incident with the lines of type (b) that it contains and the unique line of type (a) that it contains, the point ${(\infty)}$ is incident with each line of type (b).

Payne and Thas showed that this construction yields a TGQ of order ${(q^n,q^m)}$: the subgroup of ${\mathrm{PGL}(2n+m+1,q)}$ of order ${q^{2n+m}}$ that fixes ${H}$ pointwise fixes each line incident with ${(\infty)}$ and acts regularly on the set of points of ${A\backslash H}$, that is the set of points not collinear with ${(\infty)}$.

Note that when ${\mathcal{E}}$ is an oval or an ovoid then the GQ that we obtain is just ${T_2(\mathcal{E})}$ and ${T_3(\mathcal{E})}$ respectively.

In fact, every TGQ can be constructed in this manner from an egg. As mentioned previously, a TGQ is of order ${(q^n,q^m)}$ for some prime power ${q}$ and integers ${n}$ and ${m}$, and the translation group ${T}$ of the TGQ is elementary abelian of order ${q^{2n+m}}$. It is then possible to construct the associated egg in ${T}$.

If we take the dual of a known TGQ with ${2n=m}$, (that is interchange the roles of points and lines) we obtain a flock GQ. Hence we haven’t found any new GQs from this construction.

Bader, Lunardon and Pinneri proved that if ${\mathcal{E}_1}$ and ${\mathcal{E}_2}$ are eggs then there is an isomorphism from ${T(\mathcal{E}_1)}$ to ${T(\mathcal{E}_2)}$ mapping ${(\infty)}$ to ${(\infty)}$ if and only if there is a collineation of ${\mathrm{PG}(2n+m-1,q)}$ mapping ${\mathcal{E}_1}$ to ${\mathcal{E}_2}$.

Given an egg ${\mathcal{E}}$ with associated TGQ ${T(\mathcal{E})}$ we also have the dual egg ${\mathcal{E}^D}$ and so another TGQ ${T(\mathcal{E}^D)}$. This new TGQ is referred to as the translation dual of ${T(\mathcal{E})}$. This dual is different from the usual dual of a GQ formed by interchanging the roles of points and lines. When ${\mathcal{E}}$ is classical, ${T(\mathcal{E})}$ is isomorphic to its translation dual.

In the case of the Kantor pseudo-ovoids, the resulting TGQ is isomorphic to its translation dual. However, for the Cohen-Ganley and the Penttila-Williams pseudo-ovoids, the resulting TGQ is not isomorphic to is translation dual. The TGQ’s arising as the translation dual of a TGQ coming from the Cohen-Ganley pseudo-ovoid are called the Roman GQs. These translation duals are GQs that we have not encountered so far.

I have now covered all the known thick generalised quadrangles and so the series can finally come to an end.