# how to understand coxeter groups geometrically

I keep reading in the literature "Let $X$ be a Coxeter Group" but I can't think of any examples. I know they arise as Weyl groups, there are affine-Weyl groups ones as well. This list is not exhaustive.

What about examples from Kac Moody algebras. How do I understand those geometrically?

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I'm not sure I understand how to reconcile "I can't think of any examples" with "Weyl groups." – Qiaochu Yuan Oct 25 '11 at 17:04
don't laugh ... – john mangual Oct 25 '11 at 19:41
For an alternative geometric point of view, I think it's worth mentioning the Davis--Moussong complex. See Mike Davis' book The Geometry and Topology of Coxeter Groups for details. – HJRW Oct 25 '11 at 20:40

See Humphreys, starting from II.5.3. Given a Coxeter system $(W, S)$ let $V$ be the free vector space on symbols $\alpha_s, s \in S$. We define a bilinear form on $V$ by extending

$$B(\alpha_s, \alpha_t) = - \cos \frac{\pi}{m(s, t)}$$

linearly, and we define a canonical linear representation $\sigma : W \to \text{GL}(V)$ respecting $B$ by extending

$$\sigma_s \lambda = \lambda - 2 B(\alpha_s, \lambda) \alpha_s$$

multiplicatively. This representation is faithful (II.5.4), so we can study $W$ using it. The most familiar case is when $B$ is positive-definite, and this occurs if and only if $W$ is finite (II.6.4). Cases where $B$ is positive-semidefinite include the affine Weyl group case, and in general what kind of geometry we get depends on the signature of $B$ (see for example II.6.8).

Alternately, the geometric pictures are perhaps clearest for Coxeter systems of rank $3$, where the corresponding groups are essentially the triangle groups. These can be understood as symmetries of tilings of the sphere, the Euclidean plane, or the hyperbolic plane by triangles depending on the signature of $B$. This picture should be compatible with the one above (I think one just needs to consider the action of $W$ on the "unit ball" in $V$) but I haven't checked the details.

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Your statement about the unit ball is correct modulo the fact that in the Euclidean and hyperbolic cases, the unit ball has two connected components, and you should only look at one of them (you can see that all the reflections preserve the components, since they fix the hyperplane where $B(\alpha_s)$ vanishes). – Ben Webster Oct 25 '11 at 21:27

You will find pretty much everything you need in Ken Brown's Buildings: Theory and Applications (with coauther Peter Abramenko). It is all about coxeter groups/complexes and its use in the geometric theory of Buildings. For instance, chapter 2 (which is entitled 'Coxeter Groups') talks about the Canonical Linear Representation (given in the other answer on this thread), chapter 10 talks specifically about Euclidean and Hyperbolic coxeter groups, and chapter 8 contains a section on groups of Kac-Moody type. The book itself seems to be the bible of Buildings, but I may be biased since Ken himself introduced me to it.

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