Cheryl Praeger wins W.A. Scientist of the Year

News has just come through that Cheryl Praeger has been named as the 2009 Western Australian Scientist of the Year. See here for more details. Congratulations from all of us to Cheryl for her amazing contribution to pure mathematics in Australia.

Scientist of the Year

I’ve just got back from the Western Australian Science awards luncheon, and am delighted to report that Cheryl Praeger won the award, and is now officially Western Australian Scientist of the Year for 2009.

It’s quite rare for mathematics to be acknowledged in this way in direct competition with much “sexier” fields such as nanobiotechnology, and in her speech Cheryl emphasized the integral role that mathematics plays in all the sciences, and pointed out that Australia needs to lift its performance in this area given that the percentage of graduates majoring in Maths iss less than half the OECD average.

Congratulations, Cheryl!

Knarr’s model of flock quadrangles

Recently I’ve been working with generalised quadrangles which arise from flocks, and in particular, I use the model introduced by Norbert Knarr in his 1992 paper “A geometric construction of generalized quadrangles from polar spaces of rank three”. However, I’m not going to tell you everything because it would be very long. I’m not going to tell you what a generalised quadrangle is, what a flock is or even why we ought to care about such objects. Instead, my purpose is to de-mystify the Knarr model by explaining where it comes from for the classical object, the simplest case. So really, this post is intended for someone who is already familiar with the topic, but maybe later I or someone else will trace backwards to where all this began.

The three-dimensional finite Hermitian variety can be constructed in the following way. Consider the following Hermitian form on a four-dimensional vector space over :

Let the points be the one-dimensional subspaces for which this form restricts to the zero form, and let the lines be the two-dimensional subspaces for which this form computes only zero on them. Then this geometry of points and lines gives us a generalised quadrangle, and it is the classical example in the category of flock quadrangles. In other words, this geometry is a classical polar space of rank 2. Now the vectors of the vector space are certainly in one-to-one correspondence with the vectors of , but in fact, more can be said. We can define a bilinear form over by , where is some element of such that , and we see that is alternating since

.

So this form defines a symplectic space on . What is interesting in this correspondence is that the points of go to lines of , and the lines of go to solids of . Now take a point P of . Then P maps to a line P’ of . We then take an arbitrary point X on this line P’ and note that P’ is contained in . Now we project to the quotient polar space (which is isomorphic to ) via the map . So we obtain a map from totally isotropic subspaces of to totally isotropic subspaces of , and something interesting happens with respect to the point P that we started with:

	a point R of W(5,q)
lines on	a set of t.i. planes on R such that the projection to yields a BLT-set of lines of W(3,q)
lines not on	t.i. planes which meet some in a line not on R
points collinear with	lines of W(5,q) contained in some
points not-collinear with	points of W(5,q) not in the perp of R.

This is the Knarr model of a flock generalised quadrangle, where the input is a BLT-set of lines of (not just the one obtained from a pencil of lines of ). So essentially the Knarr model of a flock generalised quadrangle is the generalisation of the field reduction of , where we change the BLT-set in the resulting geometry of .

Buildings, geometries and algebraic study group VII

Alice led the discussion again this week.  At the end of the session we realised that this will be the last study group for the year due to various people going away for holidays/conferences/meetings in the lead up to christmas and of course the school’s christmas party. We will resume sometime early in the new year.

We first began by discussing the comments on the post on last week’s study group.

Recall from last week how given a building with apartments isomorphic to the Coxeter complex , we can define the W-distance on chambers. We look again at the chamber system we obtain from the Fano plane.

Let denote differ by a point and denote differ by a line. Let x be the chamber whose line is the orange line and point is the bottom left hand corner of the diagram and y be the chamber whose line is the red line and point is the top of the diagram. Then . There are four chambers which are neighbours of y. These are , which has line the red line and point the point in the middle of the line, which has line the pink line and point the top point,  which has line the orange line and point the top point, and which has line the red line and point the bottom right hand corner.  Then and . Also and while .  Moroever, . Thus we have the property that if is a chamber such that then and if ,  (where is the length of a reduced expression for g in terms of elements of S), then .

This motivates the following definition of a building: A building of type is a pair with C a set (whose elements are called chambers) and , such that for with the following hold:

• if and only if
• If is a chamber such that then and if , then .
• If then there exists such that and .

Buildings, geometries and algebraic groups study group VI

This week Alice is leading the discussion and the topic is buildings.

We begin with an example. Let and . Also let V be a 4-dimensional vector space with basis . We can form a tetrahedron with vertices labelled by . We place vertices at the midpoint of each edge (these each represent the 2-dimensional subspace spanned by the two 1-spaces at either end of the edge) and a vertex at the midpoint of each face (these represent the 3-spaces spanned by the three 1-spaces at each corner of the face). This gives us a simplicial complex whose 0-simplices are the vertices, the 1-simplices are the lines between 0-simplices, and the 2-simplices are the parts of faces surrounded by the 1-simplices. Now W acts on V by permuting the basis elements. Let I be the 2-simplex surrounded by the vertices . We can associate each of these three vertices with the subgroup of W generated by the two elements of S which fix the associated subspace. That is, is associated with , is associated with   and is associated with . The 1-simplices joining these vertices are associated with the intersections of the subgroups associated with the end vertices, while we associate I with the identity subgroup. Now for each other simplex we can associate a coset of the appropriate simplex in I.

In general, given a Coxeter system we can form the Coxeter complex which is the poset on the set and if and only if . There was much discussion as to whether or not S needs to be a simple system but this wasn’t resolved.

A building is a simplicial complex which can be expressed as a union of subcomplexes , called apartments, such that

• (B0) each apartment is a Coxeter complex;
• (B1) for any pair of simplices there exists an apartment   containing both;
• (B2) If both contain simplices then there is an isomorphism fixing and pointwise.

Taking and applying (B2) implies that all apartments are isomorphic.

Projective Planes I : PG(2,q)

This is an elementary description of the finite desarguesian projective plane and its automorphism group . I needed to explain this to a non-combinatorialist, so thought I would just add it to the blog. The required background is just elementary linear algebra, elementary group theory and the concepts of a finite field and an incidence structure.

We start with the finite field where is necessarily some power of a prime . An automorphism of a field is a permutation of the field elements such that

and the collection of all automorphisms forms a group. The automorphism group of is the cyclic group of order generated by the automorphism .

Next we construct the three-dimensional vector space with vectors being triples of elements of which we shall view as row-vectors. If is an invertible matrix with entries in , then the map is a permutation of (fixing ). The collection of all such invertible matrices forms a group called the general linear group and denoted . We can build a matrix in by picking an arbitrary non-zero vector for the first row, then choosing any vector that is not a multiple of for the second row and then any vector not in the span of for the third row. Therefore the order of the general linear group is given by

Buildings, geometries and algebraic groups study group V

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 . It has automorphism group . This can be seen by taking the points to be the edges of the complete graph and the lines to be the matchings of (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 and is the smallest thick generalised triangle. It has automorphism group . 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 .

WA Science Awards

Congratulations to Cheryl Praeger who has been shortlisted for the 2009  Western Australian Scientist of the Year.  This is for the second year running.  Cheryl leads the groups and combinatorics research group here at UWA. The awards will be presented on the 2nd of December.

Buildings, Geometries and Algebraic Groups Study Group IV

John led the discussion last week.  I am a bit late in posting as I have been busy organising the WA Junior Olympiad which was on Saturday.

The topic this week is what is called Buekenhout geometries or diagram geometries. We also have a new name.

As we have seen so far, a projective space can be associated with the Dynkin diagram of type and polar spaces can be associated with the Dynkin diagram of type . Tits’ theory of buildings generalises these notions and associates to each simple group of Lie type a geometry known as a building and this building can be encoded in the Dynkin diagram associated with the group. In his seminal 1979 paper, `Diagrams for geometries and groups’ in JCTA, Francis Buekenhout looked to generalise this further and introduced the notion of diagram geometries. These are also referred to as Buekenhout geometries.

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 and such that for every pair of adjacent (resp. non-adjacent) vertices there are (resp. ) 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 which are not incident, there is a unique line on P concurrent with .

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

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 , is a geometry of points and lines satisfying:

• every two points are on at most one line;
• each line is incident with points;
• each point is incident with lines;
• given a point P and a line which are not incident, there are lines on P concurrent with .

The point graph of such a geometry is strongly regular with and . For , 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 be a polarity of the projective space . A perp-system is a maximal set of mutually disjoint r-subspaces of 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 . By “maximal”, we mean that the cardinality of the set attains the theoretical upper bound of

The only known perp-systems are self-polar maximal arcs of , q even, and Mathon’s sporadic example in . 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 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 is a maximal subgroup of and the only known way to realise this embedding is to look to the -character table of SL(2,5):

Character table of mod GF(3).

(The above picture comes from the “The Atlas of Brauer Characters”). So we see that the representation 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 , the blow-up of a frame of :

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 . We then obtain six lines by noticing that the stabiliser of the five lines was , which is isomorphic to and there is a natural orbit of length six on lines of . 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.