On Rank 2 Complex Reflection Groups

Abstract

We describe a class of groups with the property that the finite ones among them are precisely the complex reflection groups of rank 2. This situation is reminiscent of Coxeter groups, among which the finite ones are precisely the real reflection groups. We also study braid relations between complex reflections and indicate connections to an axiomatic study of root systems and to the Shephard–Todd “collineation groups.”

Key Words:

• Braid relations
• Complex reflection groups
• Root systems

2000 Mathematics Subject Classification:

• 20F55
• 51F15

ACKNOWLEDGMENTS

The first author was partially supported by NSF grant DMS-0500873.

We would like to thank M. Broué and R. Pollack for helpful conversations.

Notes

Communicated by D. Easdown.