Some numeric results on root systems (Q811574)

Let \(\Phi\) be an irreducible root system. By a subsytem \(\Phi'\) of \(\Phi\), we mean that \(\Phi'\) is a subset of \(\Phi\) which itself forms a root system. Among all subsystems of \(\Phi\), the subsystems of \(\Phi\) of rank 2 and of type \(\neq A_ 1\times A_ 1\) are of particular importance in the theory of Weyl groups and affine Weyl groups. In the present paper, the author computes the number of such subsystems of \(\Phi\) for an irreducible root system \(\Phi\) of any type. Some interesting properties of \(\Phi\) are also obtained.