What the Heck are Coxeter Matroids?
Coxeter Matroids is a subject of mathematical research which rather nicely combines Algebra, Geometry, and Combinatorics. The following is intended for non-mathematicians (or at least for non-professional mathematicians), to give you at least some of the flavor of the subject.
The Algebra is in the form of finite Coxeter Groups, which are really finite reflection groups in Euclidean space. This means that we are dealing with symmetry groups which involve ordinary rotations and reflections. An example would be the symmetry group of a cube, in 3-space. The symmetries would include reflections parallel to the faces of the cube, reflections through diagonal planes (which contain two opposite edges of the cube), 90 degree rotations about an axis through the midpoints of two opposite faces of the cube, 180 degree rotations about an axis through the midpoints of two opposite edges of the cube, and 120 degree rotations about a diagonal axis through two opposite vertices of the cube.
The Geometry is in the form of polyhedra, such as the cube in the paragraph above, or their higher dimensional analogues, called polytopes. Here is an example of how to get some more interesting polytopes: take each edge of the cube, and take the points which are 1/4 and 3/4 of the way along that edge. Since the cube has 12 edges, we now have 24 points. Now choose just some of these 24 points, and throw the rest away (and also throw away the rest of the cube, but be careful to keep the remaining points in their original locations). Now take the "convex hull" of the remaining points. You can think of this as just taking all line segments between pairs of your points, and filling it out to get a solid object. Depending on how you picked the points you kept, you may get a Coxeter Matroid Polytope. The requirement is that any exterior edge of the polytope you just constructed must join two points which are mirror images in one of the reflections mentioned in the above paragraph.
The Combinatorics is a bit more difficult to describe. Combinatorics means the study of finite sets or families of sets, and their combinations and relationships. Matroids are a branch of combinatorics which can be viewed in one way as finite geometric configurations, and in another way as structures which allow the extremely efficient algorithmic solution of optimization problems. I'll briefly describe the latter. Suppose you want to set up a communications network between 100 different computer nodes, and you have lots of bandwidth, so all you need is that any two computers are connected by some path, no matter how lengthy, but you are only going to run links from computer to computer (no central server is involved). You are assumed to know the cost it takes to link up any two computers, and the question is how to find the most cost-efficient network possible. In this (somewhat contrived) setting, the solution is actually quite simple: pick the cheapest link, then the next cheapest, etc., except never pick a link which completes a circuit with links you have already picked. When you have picked the 99 links as cheaply as possible with no circuits completed, you will find that all 100 computers are now connected up in one network which is as cheap as possible. In other words, you never have to backtrack and change your mind about a link you chose earlier, but rather a sort of mindless "greedy" algorithm works every time. This type of structure, in which the "greedy" algorithm always works, is called a matroid.
The matroids connect with the algebra and geometry above, but in a way that is not simple to describe, so we better leave it at that. We actually get some variants of the greedy algorithm out of this approach, but they seem not to be very practical. But such algorithms are not the point of this research. The real point is the interplay of the algebra, geometry, and combinatorics, and the interesting connections it has with several other areas of higher mathematics, which, unfortunately, I cannot describe in these few paragraphs.