Unitary reflection groups associated with cyclically-invariant functions
by
Victor V. Goryunov
University of Liverpool, UK / Moscow Aviation Institute

Finite groups generated by Euclidean reflections became a very common object in various problems of singularity theory since their importance in classification of critical points of functions was demonstrated by Arnold in 1972 and 1978.

We show that a number of finite groups generated by unitary reflections are also naturally related to function singularities, namely to those invariant under a unitary reflection of finite order. To establish this one has to consider function-germs on a manifold with boundary, lift them to a cyclic covering of the manifold ramified over the boundary and study the monodromy of the symmetric Milnor fibres. The construction provides a new notion of roots for the groups under consideration and skew-Hermitian versions of these groups.

The work can be considered as an initial step in solving a problem of finding singularity theory interpretations of finite groups generated by unitary reflections which was posed by Arnold some twenty years ago.

Date received: May 27, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacy-14.