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 1as 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 2Any basis can be written as for some orthogonal to no element of .

Next, there is another result:

Proposition 3Given , 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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