## 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

Neat!

Comment by Jonathan Vos Post | February 4, 2010 |

[...] we’re ready to describe the action of the Weyl group of a root system on the set of its Weyl chambers. Specifically, the action is “simply transitive”, and the group itself is generated by [...]

Pingback by The Action of the Weyl Group on Weyl Chambers « The Unapologetic Mathematician | February 5, 2010 |

[...] Fundamental Weyl Chamber When we first discussed Weyl chambers, we defined the fundamental Weyl chamber associated to a base as the collection of all the [...]

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