If you define a Coxeter group as one with a finite presentation [x_j | x_j^2 = (x_j x_k)^(m_jk) = 1, 1 <= j,k <= n] (where m_jk is allowed to be oo, meaning x_j x_k is infinite order) then I understand that sometimes this group cannot be represented as a reflection group. --Dan On 2012-11-29, at 5:01 AM, Veit Elser wrote:
On Nov 29, 2012, at 1:56 AM, Mike Stay <metaweta@gmail.com> wrote:
On Wed, Nov 28, 2012 at 11:12 AM, Michael Kleber <michael.kleber@gmail.com> wrote:
The canonical representation works by defining the basis vectors to have whatever inner products you want them to have in order to satisfy the Coxeter relationships you're given. You end up with a vector space and a metric, but the metric is presumably not going to have any nice properties.
So, I suppose I'd have to do something involving the Gram-Schmidt procedure to get an orthonormal basis in order to encode the reflections into matrices...
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
You get the reflection-vectors directly from the Cholesky decomposition of their Gram matrix (matrix of inner products). Since the latter has only rational elements in the case of Coxeter groups, the Cholesky vectors will at worst have square roots. Forming reflection matrices from the vectors is of course automatic.