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 ≠A1× A1 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.
