So, suppose given a root system in a euclidean space
, which arises from a semisimple Lie algebra and a Cartan subalgebra as before. The first goal of this post is to discuss the “splitting”
(disjoint union) in a particular way, into positive and negative roots, and the basis decomposition into simple roots. Here .
To do this, choose such that
for
. Then define
to be those roots
with
and
those with
. This was easy. We talked about positive and negative roots before using a real-valued linear functional, which here is given by an inner product anyway.
Bases
OK. Next, I claim it is possible to choose a linearly independent set such that every root is a combination
with all the or all the
.
Then will be called a base. It is not unique, but I will show how to construct this below.
An amusing application of the condition on the simple roots is that when
; otherwise, if
we would have
but this has occurring with coefficient
and
occuring with negative coefficient. This contradicts the hypothesis on simple roots.
Anyway, to construct , first choose
as above and the corresponding splitting
. Let
denote the set of positive roots that cannot be written as the sum of two positive roots—in other words, the indecomposable ones. I claim
spans
. Indeed, it is enough to check that each positive root is a sum of indecomposable ones. If
is not, we can choose
such that
is minimal among roots which are not sums of indecomposable ones; then, however,
is itself not indecomposable, so we can write
for
. Then
are by the inductive hypothesis (since
) sums of indecomposable ones; thus so is
, contradiction.
Next, I claim that as constructed above via indecomposable roots satisfies
for distinct
, which we showed was necessary for a construction of simple roots. If
then
is a root (cf. the discussion of maximal strings here). If
, then
is not indecomposable. Similarly if
, then
is not indecomposable. This will let us prove linear independence. Indeed, if we had an expression
for distinct sets and
, then taking inner products with itself yields
which is a contradiction by unless all the
are zero, which is impossible since taking inner products with
gives something positive.
Incidentally, this resembles the argument in Sylvester’s theorem on quadratic forms.
Proposition 1
as just defined is a base.
We will denote it by to emphasize the dependence on
.
It is in fact the case that any base can be obtained in such a fashion. To see this, we will construct
such that
for all
(and consequently
is orthogonal to no member of
). I claim then that
. Indeed,
leads to a decomposition
where
consists of roots that are positive linear combinations of
, and
those that are negative linear combinations of
. It is clear that the splitting
is also obtained by taking inner products with
as at the beginning of this post—i.e.,
consists of those roots that bracket positively with
. Now it is clear that
is indecomposable because it is a basis, and the set of indecomposable elements of
is itself a basis by the above arguments of Proposition 1. If a basis is contained in another, the two are equal though. So
.
It now remains to construct from
, and in fact we can do it for any basis of
. For each
, one defines
to be the projection of
into the space spanned by
. Then
is orthogonal to
but brackets positively with
. So
brackets positively with all of
.
Proposition 2 Any basis can be written as
for some
orthogonal to no element of
.
Next, there is another result:
Proposition 3 Given
, we can write
where each partial sum is a positive root.
We can write as a sum
where
. Induct on the degree
; if it is one, then
already. If not, then
is at least not orthogonal to all of
. Now if
for all
then it is easily checked that
would be linearly independent, so we can choose
with
As a result , and it is moreover a positive root. We can apply the inductive hypothesis to
and use
.
Weyl chambers
Consider ; this is the complement of the union of several hyperplanes. Each
corresponds to a basis
, and it is clear that
if
are sufficiently close to one another. In particular, the connected components of
are in one-to-one correspondence with the bases of
.
These connected components are called Weyl chambers. Next, we’ll define a group that permutes the Weyl chambers.
I learned this from Shlomo Sternberg’s book.
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