Some Numeric Results on Root Systems

Let Φ be an irreducible root system (sometimes we denote Φ by Φ(X) to indicate its type X). Choose a simple root system Π in Φ. Let Φ + (resp. Φ~) be the corresponding positive (resp. negative) root system of Φ. By a subsystem Φ' of Φ (resp. of Φ + ) , we mean that Φ' is a subset of Φ (resp. of Φ) which itself forms a root system (resp. a positive root system). We refer the readers to Bourbaki's book for the detailed information about root systems. Among all subsystems of Φ, the subsystems of Φ of rank 2 and of type Φ A\ x A\ are of particular importance in the theory of Weyl groups and affine Weyl groups (see the papers by Jian-yi Shi). In the present paper, we shall compute the number of such subsystems of Φ for an irreducible root system Φ of any type. Some interesting properties of Φ are also obtained.