Root system

Definition
A root system is a finite set of nonzero vectors in $$\R^n$$ called roots satisfying the following conditions:


 * The roots span $$\R^n$$
 * For any root $$\alpha$$, $$\alpha$$ and $$-\alpha$$ are precisely the multiples of $$\alpha$$ that belong to the root system
 * For any roots $$\alpha,\beta$$ the reflection of $$\alpha$$ about the hyperplane perpendicular to $$\beta$$, is also a root
 * For any two roots $$\alpha,\beta$$, the ratio of inner products:

$$n_{\beta\alpha} = \frac{(\beta,\alpha)}{(\alpha,\alpha)}$$

is an integer.

The rank of a root system is defined as the dimension of the vector space it spans.

Positive roots
Any hyperplane divides the vector space into two parts. By calling one of them the positive part, we can distinguish roots into two types: the positive roots and the negative roots. Note that for every root, either that or its negative, is a positive root.

Different choices of hyperplane (with directed normal) may end up giving the same set of positive roots. The set of all hyperplanes giving a particular set of positive roots is termed a Weyl chamber for the root system. Clearly, there are only finitely many Weyl chambers.

Simple roots
Relative to a choice of positive roots, a simple rooot is a positive root that cannot be expressed as a sum of positive roots. Using the properties of root systems, we can show that:


 * Every system of positive roots gives a corresponding system of simple roots
 * Two different systems of positive roots cannot give the same system of simple roots
 * A system of simple roots forms a basis for the whole vector space

Dynkin diagram
The Dynkin diagram of a root system is a graph that stores all geometric information from which the root system can be constructed. Because of the fact that every root system can in fact be constructed (upto scaling) from a knowledge of the angles between all the pairs of roots.