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
February 3, 2010
Posted by John Armstrong |
Geometry, Root Systems |
3 Comments
