John led the discussion again this week and looked at coset geometries and polar spaces.

John began with the following pictureThis is an incidence geometry with 15 points and 15 lines each containing 3 points. It is the smallest thick generalised quadrangle and denoted by $W(3,2)$. It has automorphism group $S_6$. This can be seen by taking the points to be the edges of the complete graph $K_6$ and the lines to be the matchings of $K_6$ (that is the sets of three disjoint edges). The incidence graph of this geometry is known as Tutte’s 8-cage or the Tutte-Coxeter graph.

John next drew the Fano planeThis projective plane is denoted by $PG(3,2)$ and is the smallest thick generalised triangle. It has automorphism group $PGL(3,2)\cong PSL(2,7)$. The incidence graph of this geometry is the Heawood graph.

Next we had the smallest (thick) generalised digon, which is the geometry with three points and three lines such that each line consists of all three points. The incidence graph for this geometry is the complete bipartite graph $K_{3,3}$.

Now let $G=A_7$, the alternating group on 7 letters and define three subgroups $G_1=A_6$, $G_2=(S_3\times S_4)\cap A_7$ and $G_3=PSL(3,2)$ of G. The Neumaier geometry is the geometry $\Gamma$ whose points are the right cosets of $G_1$ in $G$, lines are the right cosets of $G_2$ in $G$ and planes are the right cosets of $G_3$ in $G$.  For $i\neq j$ we define $G_ix$ to be incident with $G_jy$ if and only if $G_ix\cap G_jy\neq \emptyset$, or equivalently, $xy^{-1}\in G_iG_j$.  There are 7 points, 35 lines and 15 planes. There is not meant to be any “geometrical” implications by defining elements to be points, lines or planes, we could have equally defined the elements of our geomery to be elements of Type 1, Type 2 or Type 3. Note that G acts as a group of automorphism of $\Gamma$ by right multiplication.  This action is flag-transitive, that is, given any two flags $F_1,F_2$ of the same type, there is an element of G which maps $F_1$ to $F_2$.

This is an example of a coset geometry. Given a group G and subgroups $G_1,G_2,\ldots,G_n$ we can define a rank n pregeometry whose elements of type 1 are the right cosets of  $G_1$ in G, elements of type 2 are the right cosets of $G_2$ in G and so on, so that the elements of type n are the right cosets of $G_n$ in G. There are conditions on the $G_i$ needed so that this pregeometry is a geometry. Any flag-transitive geometry can be constructed as a coset geometry, by taking G to be the full automorphism group, choosing a maximal  flag $F=\{\alpha_1,\ldots,\alpha_n\}$ and for $i\in\{1,\ldots,n\}$ letting $G_i$ to be the stabiliser in G on $\alpha_i$.

Returning to the Neumaier geometry,  the residue of the point corresponding to the coset $G_1$ is the coset geometry for $G_1\cong A_6$ using the subgroups $G_1\cap G_2$ and $G_1\cap G_3$. Now $G_1\cap G_2\cong S_4$ and $G_1\cap G_3\cong S_4$ and so this residue has 15 lines and 15 planes. This is the same subgroup structure given by $A_6$ acting on the generalised quadrangle $W(3,2)$ seen at the beginning of this post.  Hence the residue is isomorphic to $W(3,2)$.

The residue of the line corresponding to the coset $G_2$ is the coset geometry for $G_2$ using the subgroups $G_2\cap G_1\cong S_4$ and $G_2\cap G_3\cong S_4$. This residue has 3 points and 3 planes and is the generalised digon seen earlier.

Finally the residue of the plane corresponding to the coset $G_3$ is the coset geometry for $G_3\cong PSL(3,2)$ using the subgroups $G_3\cap G_1\cong S_4$ and $G_3\cap G_2\cong S_4$. This residue contains 7 points and 7  lines and is isomorphic to the Fano plane. Hence the diagram for the Neumaier geometry is the $C_3$ diagram.

[added 9/1//09]: John has provided the subgroup lattice for $A_7$ displaying $G_1,G_2,G_3$ and their intersections.

Finite Polar Spaces

Next we looked at polar spaces. These are best introduced by way of example. We saw in a previous week that we can form a geometry by taking an alternating form on a vector space V and taking the elements of the geometry to be the totally isotropic subspaces (that is, those for which the restriction of the form is 0) and incidence is symmetrised inclusion.

More generally, let V be a d-dimensional vector space over GF(q). Let $\sigma$ be an automorphism of GF(q) and B be a $\sigma$-sesquilinear form on V, that is, B satisfies

• $B(v_1+v_2,w)=B(v_1,w)+B(v_2,w)$
• $B(v,\alpha w)=\alpha^\sigma B(v,w)$
• $B(\alpha v,w)=\alpha B(v,w)$

for all $v,w,v_1,v_2\in V$ and $\alpha\in GF(q)$. We also require that B is reflexive, that is $B(v,w)=0$ implies $B(w,v)=0$. The totally isotropic subspaces are those for which the restriction of B is 0.

A duality of the projective space $PG(d-1,q)$ is an inclusion reversing permutation of the subspaces. A polarity is a duality of order 2.  Our sesquilinear form defines a polarity by mapping each subspace U to the subspace $U^\perp=\{v\in V\mid B(v,u)=0 \text{ for all } u\in U\}$.

In fact, every polarity arises from a reflexive sesquilinear form as follows: Given a polarity $\pi$ of a projective space with defining vector space V,  if X is a subset of  V, we let $X^\circ$ be the annihilator of X, that is, those functionals in $V^*$ which compute 0 on all elements of X. Now we compose $\pi$ with the map $x\mapsto x^\circ$ and obtain a collineation of the projective space. Hence there is a semilinear map f with associated field automorphism $\sigma$ such that $\pi\circ = f$. In other words, $\pi (X)=\{ v\in V\mid f(x)v=0 \text{ for all } x \in X\}$. Thus we obtain a nondegenerate sesquilinear form given by $B(u,v)=f(v)u.$

Given our sesquilinear form we can define a geometry whose elements are the totally isotropic subspaces and incidence is given by symmetrised inclusion. Such geometries are called polar spaces.

Another way of constructing a polar space is to take a quadratic form Q on V. This is a map from V to GF(q) such that $Q(\lambda v)=\lambda^2 Q(v)$ and there is a symmetric bilinear form B such that $Q(v+w)=Q(v)+Q(w)+B(v,w)$. Totally singular subspaces are those for which the restriction of Q is 0. We then get a polar space whose elements are the totally singular subspaces.

Following Buekenhout and Shult, a polar space can be defined axiomatically as  a partial linear space such that

• no point is collinear with every point
• there are at least 3 points on every line
• given a point P and a line $\ell$, either P is collinear with all points on $\ell$ or P is collinear with a unique point on $\ell$.

We can then define subspaces of rank greater than 2 in such a polar space by taking sets of mutually collinear points which are a union of lines.

Tits then proved that all finite polar spaces of rank at least 3 are classical, that is, arise from a vector space equipped with a sesquilinear of quadratic form. This is analogous to the Veblen-Young axioms for projective spaces which define a projective space axiomatically in terms of points and lines and the only projective spaces of rank at least 3 are the ones obtained by taking the subspaces of a vector space over a division ring.