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.

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