Properties of Irreducible Root Systems II
We continue with our series of lemmas on irreducible root systems.
If is irreducible, then the Weyl group
acts irreducibly on
. That is, we cannot decompose the representation of
on
as the direct sum of two other representations. Even more explicitly, we cannot write
for two nontrivial subspaces
and
with each one of these subspaces invariant under
. If
is an invariant subspace, then the orthogonal complement
will also be invariant. This is a basic fact about the representation theory of finite groups, which I will simply quote for now, since I haven’t covered that in detail. Thus my assertion is that if
is an invariant subspace under
, then it is either trivial or the whole of
.
For any root , either
or
. Indeed, since
, we must have
. As a reflection,
breaks
into a one-dimensional eigenspace with eigenvalue
and another complementary eigenspace with eigenvalue
. If
contains the
-eigenspace, then
. If not, then
is perpendicular to
or
couldn’t be invariant under
, and in this case
.
So then if isn’t in
then it must be in the orthogonal complement
. Thus every root is either in
or in
, and this gives us an orthogonal decomposition of the root system. But since
is irreducible, one or the other of these collections must be empty, and thus
must be either trivial or the whole of
.
Even better, the span of the -orbit of any root
spans
. Indeed, the subspace spanned by roots of the form
is invariant under the action of
, and so since
is irreducible it must be either trivial (clearly impossible) or the whole of
.
Are we ready for Dynkin diagrams?
Not yet.
[…] and be two roots. We just saw that the -orbit of spans , and so not all the can be perpendicular to . From what we discovered […]
Pingback by Properties of Irreducible Root Systems III « The Unapologetic Mathematician | February 12, 2010 |
[…] also see that the Weyl orbit of a root spans the plane in the irreducible cases. But, again, in the Weyl orbits of and only span their […]
Pingback by Some Root Systems and Weyl Orbits « The Unapologetic Mathematician | February 15, 2010 |