Some Combinatorial Results for Complex Reflection Groups

In this paper, we prove that a simple system for a subsystem Ψ of the complex root system Φ can always be chosen as a subset of the positive system Φ+of Φ. Furthermore, we show that a set of distinguished coset representatives can be found for every reflection subgroup of the complex reflection groups. The corresponding results for real crystallographic root systems and their reflection groups (i.e., Weyl groups) are well known (see [9]).