Summary: ON RANK-TWO COMPLEX REFLECTION GROUPS
PRAMOD N. ACHAR AND ANNE-MARIE AUBERT
Abstract. We describe a class of groups with the property that the finite
ones among them are precisely the complex reflection groups of rank two.
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."
A complex reflection group is a finite group of transformations of a complex vector
space generated by complex reflections or pseudo-reflections, i.e., transformations
that fix some hyperplane. Any finite Coxeter group can naturally be thought of
as a complex reflection group, simply by complexifying the vector space on which
the reflection representation acts, but there are many complex reflection groups
that do not arise in this way. Recent work by a number of people has shown that
various structures attached to Weyl groups, can be generalized to complex reflection
groups, even though there is no analogue of the underlying algebraic group.
Many aspects of the theory of finite Coxeter groups are actually present in the
much broader setting of all Coxeter groups. Indeed, it should be remembered that
the following characterization of reflection groups is a theorem, not a definition: