Hall of mirrors: Coxeter Groups and the Davis Complex

I’ve spent a lot of time this summer thinking about the best way to present maths. When someone gives a talk or lecture they normally write on the board in chalk or present via Beamer slides. Occasionally though, someone comes along with some great hand drawn slides that make listening to a talk that wee bit more exciting. So the images in this blog are part of my tribute to this new idea.

I’ve talked about Coxeter groups before (here), but I’ll start afresh for this post. It is worth mentioning now that Coxeter groups arise across maths, in areas such as combinatorics, geometric group theory, Lie theory etc. as well as topology.

A Coxeter group is a group generated by reflections, and “braid type” relations. Informally, you can imagine yourself standing in the middle of a room with lots of mirrors around you, angled differently. Your reflection in a single mirror can be viewed as a generator of the group, and any other reflection through a series of mirrors can be viewed as a word in the group. Here is a silly picture to show this:

silly picture

Formally, a Coxeter group is defined by a Coxeter matrix on the generating set S. This is an S by S matrix, with one entry for each ordered pair in the generating set. This entry has to be 1 on the diagonal of the matrix or a whole number that is either bigger than 2 or infinity (\infty ) off the diagonal. It also has to be symmetric, having the same entry for (t,s) as (s,t) . See below:

coxeter groups1

Given this matrix you can then define the Coxeter group to be the group generated by S with relations given by the corresponding entry in the matrix.

coxeter groups2

In particular notice that the diagonal of the matrix being 1 gives that each generator squares to the identity, i.e. it is an involution. It can be hard to see what is going on in the matrix so there is a nicer way to display this information: a Coxeter diagram. This is a graph with a vertex for every generator, and edges which tell you the relations, as described below:

coxeter groups3

The relation (st)^{m_{st}} can also be rewritten as ststs...=tstst... where there are m_{st} elements on each side. This is reminiscent of the braid relations, hence why I called them “braid like” before. In the mirror analogy, this says the mirrors are angled towards each other in such a way that you get the same reflection by bouncing between them m times, independent of which of the two mirrors you turn towards to begin with.

There exist both finite and infinite Coxeter groups. Here is an example of a finite Coxeter group, with two generators s and t. If you view them as reflections on a hexagon (as drawn) then doing first s and then t gives a rotation of 120 degrees, and so doing st 3 times gives the identity, as required.

coxeter example

On the other hand, if you add another generator u with a braid-3 relation with both s and t , then the group is infinite. You can imagine tiling the infinite plane with triangles. If you take s , t and u to be reflections in the 3 sides of one of these triangles then they satisfy the relations they need to, and you can use these three reflections to transport the central triangle to any other one. If you think about this for a while, this shows the group is infinite. A (somewhat truncated) picture is shown below.

coxeter example 1

Examples of Coxeter groups don’t just live in 2-D Euclidean space. There is another finite group which acts as reflections on the permutahedron:

coxeter-complex

And other Coxeter groups which act as reflections on the hyperbolic plane.

The mathematical object I am working with at the moment is called the Davis Complex. You can build it out of finite subgroups of a Coxeter group (side note for the mathematicians: taking cosets of finite subgroups and forming a poset which you can then realise). Even infinite Coxeter groups have lots of finite subgroups. The great thing about the Davis complex being built out of finite things is that there is a classification of finite Coxeter groups! What this means is that when you have a finite Coxeter group its diagram either looks like one of the diagrams below, or a disjoint collection of them.

classification of finite coxeter

So because we only have a few diagrams to look at in the finite case, we can prove some things! Right now I am working on some formulas for staring at the Coxeter diagrams and working out the homology of the group. I’m using the Davis complex and all its nice properties to do this. I’ll leave you with a picture of the Davis complex for our first example.

davis example 2

davis example

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