## Weyl Chambers

A very useful concept in our study of root systems will be that of a Weyl chamber. As we showed at the beginning of last time, the hyperplanes for cannot fill up all of . What’s left over they chop into a bunch of connected components, which we call Weyl chambers. Thus every regular vector belongs to exactly one of these Weyl chambers, denoted .

Saying that two vectors share a Weyl chamber — that — tells us that and lie on the same side of each and every hyperplane for . That is, and are either both positive or both negative. So this means that , and thus the induced bases are equal: . We see, then, that we have a natural bijection between the Weyl chambers of a root system and the bases for .

We write for and call this the fundamental Weyl chamber relative to . Geometrically, is the open convex set consisting of the intersection of all the half-spaces for .

The Weyl group of shuffles Weyl chambers around. Specifically, if and is regular, then .

On the other hand, the Weyl group also sends bases of to each other. If is a base, then is another base. Indeed, since is invertible will still be a basis for . Further, for any we can write , and then use the base property of to write as a nonnegative or nonpositive integral combination of . Hitting everything with makes a nonnegative or nonpositive integral combination of , and so this is indeed a base.

And, just as we’d hope, these two actions of the Weyl group are equivalent by the bijection above. We have because preserves the inner product, and so . Thus we write for some regular and find that