Continuation of my notes from “Reflection Groups and Coxeter Groups” by James E. Humphreys.
is a simple system. Let be in and let be the reflection associated with . Of course, must take into -that’s what reflection does. But what does it (i.e.) do to the other members of (the positive system containing )? It is comforting to know that will not take any member of (except , of course) outside .
A simple reflection maps a positive system into itself but for the one root sent to its negative.
How annoying it would if this were not the case, like the contents of a neat cupboard thrown all over the room. This neatness,this organization serves as a good rule of thumb when I am thinking about stuff. If the end result is chaotic, it is highly likely I have done things wrong.
Anyway, the upshot of a positive system (but one root etc.) being reflected into itself is that any two positive systems lying within a root system are conjugate to each other under the action of the associated reflection group. Ditto for simple systems in a root system. Hence one can fix a simple system to prove results, without loss of generality.