Some more on simple systems

Continuation of my notes from “Reflection Groups and Coxeter Groups” by James E. Humphreys.

$\Delta$ is a simple system. Let $\alpha$ be in $\Delta$ and let $s_\alpha$ be the reflection associated with $\alpha$. Of course, $s_\alpha$ must take $\alpha$ into $-\alpha$-that’s what reflection does. But what does it (i.e.$s_\alpha$) do to the other members of  $\Pi$ (the positive system containing $\Delta$)? It is comforting to know that $s_\alpha$ will not take any member of $\Pi$ (except $\alpha$, of course) outside $\Pi$.

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.