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 ot