Summary: ON RANKTWO COMPLEX REFLECTION GROUPS
PRAMOD N. ACHAR AND ANNEMARIE 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 in
dicate connections to an axiomatic study of root systems and to
the ShephardTodd ``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.