On Thu, 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...
No no, that might well be impossible. The metric that comes directly from the Coxeter relationships is only going to be positive-definite if the Coxeter group is finite. Otherwise the canonical representation isn't in Euclidean space -- when you try to do Gram-Schmidt, you'll find some vectors have length-squared that's zero or negative. --Michael
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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