Here is the next instalment in our Phan systems study group which was held yesterday. I forgot to bring my laptop so I took notes by hand. Blogging the study group has already had one benefit as Gordon is now attending. Having been initially put off by the title he has realised that we are actually doing lots of stuff that he is interested in learning.
John began the discussion by working through the geometry of the symplectic polar space of rank 2. Let and let be the bilinear form on , for some field K, given by . Note we are using row vectors, and if , we have . Then is an alternating form, that is and for all . The totally isotropic subspaces of V, are those subspaces W for which the restriction is zero, that is, for all . We can then form a point-line geometry whose points are the totally isotropic 1-spaces of V and the lines are the totally isotropic 2-spaces of V. Note that all 1-spaces of V are totally isotropic, but V does not contain any totally isotropic 3-spaces. A hyperbolic pair is a pair of vectors such that .
Let and . Then and are hyperbolic pairs, while and are totally isotropic. We have , that is, V can be written as an orthogonal direct sum of two hyperbolic lines. One of the chambers in this geometry is .
The point-line geometry we have just defined is an example of a generalised quadrangle. That is, it satisfies the following two axioms:
- Any two points lie on at most one line.
- Given a line and a point p not incident with , there is a unique point q on which is incident with p.
Such a geometry is often referred to as a geometry of type and is denoted by the diagram
In general, we can define an alternating bilinear form on by using the matrix . The set of all totally isotropic subspaces of forms the symplectic polar space of rank n. The maximal totally isotropic subspaces have dimension n and can be written as an orthogonal direct sum of n hyperbolic lines. This geometry has the diagram :where the nodes of the diagram represent from left to right the totally isotropic 1-spaces, totally isotropic 2-spaces through to the totally isotropic n-spaces. Recall that the edge between nodes i and j represents the geometry formed by the residue of a flag containing elements of all types except i and j. So for example, if we take a flag where each is a totally isotropic i-space then the residue of F is the set of all totally isotropic 1-spaces and 2-spaces incident with F. Since is totally isotropic, this is all the 1-spaces and 2-spaces contained in and hence is a projective plane. This is why a single edge is placed between the nodes representing 1-spaces and 2-spaces. At the other end of the diagram, if we take a flag with each totally isotropic of dimension i, then the residue of is the set of all totally isotropic n-spaces and -spaces containing . Since has dimension and has dimension 4 and is isomorphic to our original symplectic polar space of rank 2, we have a double bond between the last two nodes in the diagram.
John mentioned that whereas a geometry with the same diagram as a projective space is indeed a projective space, a geometry with the diagram is either a symplectic polar space or the Neumaier geometry. Hopefully we learn more about this is subsequent weeks.
[added19/10/09: John has pointed out that I have misunderstood him here. The diagram is used to denote any generalised quadrangle, and a geometry with diagram is either a polar space or the Neumaier geometry. There are polar spaces other than the symplectic one with diagram . This is something we should be covering later.]
Now onto the related groups. The symplectic group is the group of all linear transformations of which preserves . This is all matrices such that . Writing A as with each a matrix we see that we must have and .
A Borel subgroup B is the stabiliser of a chamber. Using our chamber we have that B is all matrices in with lower triangular, and , the inverse-transpose of .
A frame is . Then N is the setwise stabiliser of and so is a group of monomial matrices. The group N must preserve the partition and so . The base group of this wreath product is generated by the matrices
These matrices interchange and , and and respectively. The top group is given by all matrices
where P is a permutation matrix. The kernel T of the action of N on is all diagonal matrices of the form
So the Weyl group of is .
This can all be generalised to where we have that the Weyl group is , the Coxeter group of type . This is also thought of as the group of signed permutations of the set .
To finish off, Alice spoke briefly about the apartments in this setting. Whereas I had initially interpreted her as saying that in the case an apartment was just the same as a frame, it is actually more subtle than that. In fact an apartment is the set of all flags which you can get from the elements of a frame. Thus in the case of , the apartment corresponding to is the set of all flags whose elements are in .This can be represented in the following picture where the vertices are 1-spaces and the edges are 2-spaces. Each vertex-edge incident pair is a rank 2 flag.
The chamber system of this apartment is then given below. I have only labelled three of the nodes.The red edges represent that the 1-spaces of the chamber are different while the green edges represent that the 2-spaces of the chamber are different. You can define reflections of the apartment (the square picture above) such that is the reflection about the vertical axis of the square and is the reflection about the diagonal axis from the top left corner. Then starting in the bottom left hand corner of the hexagon and working clockwise we get the sequence which maps the bottom left chamber around to the top right hand corner chamber. Working anticlockwise we get around to the top right hand corner. Since the diagram should commute we get and so , that is we get the Coxeter relation for groups of type .