I am having trouble warping my head around the exchange and deletion conditions in finite reflection groups (i.e.Coxeter groups). It is mentioned as the "characterising property of coxeter groups among groups generated by involutions".(http://qchu.wordpress.com/2010/07/08/the-strong-exchange-condition/) Further,"So the strong exchange and deletion conditions must, in principle, be enough to answer any question one could ask of an arbitrary Coxeter group."
When I follow the proof given in the text Reflection Groups and Coxeter Groups by James E. Humphreys, I notice that the following facts are used to prove it: 1.a simple reflection permutes the positive root system except for the one root which is sent to its negative, 2.a positive root can be written as a nonnegative linear combination of simlple roots.This is built into the definition of simple root systems. 3.definition of reflection.
However, I read this on http://planetmath.org/encyclopedia/WeylGroup2.html. The remaining condition to make a Weyl group a Coxeter group is the exchange condition. Thus every finite Weyl group with the exchange condition is a Coxeter group, and visa-versa.
And I am more confused.Why would a Weyl group fail to have the exchange condition, when the only thing stopping it from being a Coxeter group is the tight set of restrictions placed on the root lengths. To summarize, I am asking if someone can help me see why the deletion and exchange conditions play such an important role in "coxeterness" of a group? And why does a Weyl group fail to satisfy these conditions? Note: All terms as used in the text mentioned in the question. I am assuming this is standard.If not, please let me know.