A construction of Mathon’s perp-system

This post is a little report on some recent work of mine and Frank De Clerck (Ghent University), which we have recently submitted after sitting on it for a few years. First I will give some background. Strongly regular graphs are a bubbling topic in combinatorics, since many of your favourite graphs are strongly regular, they have nice algebraic properties (e.g., three distinct eigenvalues) and they were borne essentially from the theory of experimental designs. If that doesn’t convince you enough that they are important, well they have their own Mathematics Subject Classification code (namely 05E30). A regular graph is strongly regular if there are two constants \lambda and \mu such that for every pair of adjacent (resp. non-adjacent) vertices there are \lambda (resp. \mu) common neighbours.

There are some rank 2 finite geometries whose point-graphs are strongly regular, and these geometries are somewhat rare, and beautiful when they crop up (like pure mathematicians I guess). The point-graph of a rank 2 geometry is simply the graph you get when you take as vertices the points, and the adjacency relation induced by the collinearity relation. So for example, take a (thick) generalised quadrangle. This is a geometry of points and lines such that

  • every two points are on at most one line;
  • every line has at least three points;
  • given a point P and a line \ell which are not incident, there is a unique line on P concurrent with \ell.

From these axioms it follows that there are two constants s and t such that every line has s+1 points and every point is incident with t+1 lines. The point-graph is then a strongly regular graph of valency s(t+1) and with \lambda=s-1, \mu=t+1.

A generalised quadrangle is an example of a partial geometry, and so belongs to a wider class of geometries which yield strongly regular point graphs.  A partial geometry with parameters (s,t,\alpha), is a geometry of points and lines satisfying:

  • every two points are on at most one line;
  • each line is incident with s+1 points;
  • each point is incident with t+1 lines;
  • given a point P and a line \ell which are not incident, there are \alpha lines on P concurrent with \ell.

The point graph of such a geometry is strongly regular with \lambda = s-1+t(\alpha-1) and \mu=\alpha(t+1). For \alpha=2, there are not many known partial geometries, just (i) the Van Lint – Schrijver geometry, (ii) the Haemers geoemetry and the (iii) partial geometry of Mathon. It is the latter which we are interested in.

The Mathon partial geometry arises as the linear representation of Mathon’s perp-system. Let \rho be a polarity of the projective space PG(d,q). A perp-system is a maximal set of mutually disjoint r-subspaces of PG(d,q) such that every pair of elements from this set are mutually opposite (disjoint from the perp of the other). By “pair” here, we do not require that the two elements be distinct, so in particular, every element of a perp-system is non-singular with respect to \rho. By “maximal”, we mean that the cardinality of the set attains the theoretical upper bound of

q^{(d-2r-1)/2}(q^{(d+1)/2)}+1)/(q^{(d-2r-1)/2}+1).

The only known perp-systems are self-polar maximal arcs of PG(2,q), q even, and Mathon’s sporadic example in PG(5,3). Mathon’s example had no geometric construction, it was simply written down in coordinate form, but it was known that its stabiliser was isomorphic to S_5 and that it was a perp-system with respect to a symplectic, hyperbolic and elliptic polarity. (See the seminal paper on perp-systems by De Clerck, Delanote, Hamilton and Mathon). Now S_5 is a maximal subgroup of PGSp(6,3) and the only known way to realise this embedding is to look to the GF(3)-character table of SL(2,5):

Character table of $latex A_5$ mod GF(3).

(The above picture comes from the “The Atlas of Brauer Characters”). So we see that the representation \varphi_7 has degree 6 and is symplectic. Anyway, what we have now is a geometric reason for this embedding, a realisation of SL(2,5) as the stabiliser of a sporadic perp-system. All it really needed was a nice geometric construction, and that’s what Frank and I were able to do.

We start with a set of four lines \mathcal{F}_4, the blow-up of a frame of PG(2,9):

\ell_1=(I\,O\,O),\ell_2=(O\,I\,O),\ell_3=(O\,O\,I), \ell_4=(I\,I\,I)

where O and I are the zero and identity two-by-two matrices respectively. We then have a way of obtaining a fifth line, having the property that it is totally isotropic with respect to a particular symplectic form, and that it is disjoint from the span of any two elements of \mathcal{F}_4. We then obtain six lines by noticing that the stabiliser of the five lines was S_5, which is isomorphic to PGL(2,5) and there is a natural orbit of length six on lines of PG(5,3). Now notice that 6 choose 2 is 15. This allows us to obtain fifteen more lines, giving us 21 lines in total; this is Mathon’s perp-system! We also show that we can construct the smallest generalised quadrangle W(2) from the line sets encountered above.

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