
Project Summary

The study of symmetry in geometrical and abstract contexts is a central issue in such diverse areas as mathematical physics, singularity theory, algebraic geometry, quantum groups and the study of knots and braids. Group theory provides the mathematical framework for the analysis of symmetry. Reflection groups, simple examples of which are the symmetry groups of the five platonic solids, play a key role in all of the areas mentioned above. Thus an improved understanding of reflection groups will significantly enhance the development of several important theories.






Approach

My hope is to use the funding provided by the Australian Research
Council to support visits to Sydney by other mathematicians interested
in studying reflection groups, to collaborate with me on a variety
of related problems (see below). People interested in reflection
groups are welcome to contact me: email
bobh@maths.usyd.edu.au.
I also propose to travel to more international conferences than I
have been able to previously.






Specific objectives

Here is a sample of the kinds of problems that may be addressed.

Find methods of constructing of Wgraphs for
irreducible representations of finite Coxeter groups and their
associated IwahoriHecke algebras.

Study automorphism groups of Coxeter groups.
If an infinite irreducible Coxeter group has the property that all
of its rank 2 parabolic subgroups are finite, does it follow that
its full automorphism group is generated by inner automorphisms
and graph automorphisms?

Rigidity of Coxeter groups: can an infinite irreducible Coxeter
group with finite rank 2 parabolics have two nonconjugate
Coxeter systems?

Discover further extensions of the theory of Coxeter groups from
the Euclidean (finite) case to the nonEuclidean (infinite)
case. Hyperbolic Coxeter groups are an important special
case, of course, but my aim is not to restrict attention
to these.

In my dreams, I sometimes think that it may be possible to obtain a
theory of unitary reflection groups analogous to the theory of Coxeter
groups.

In all cases, algorithmic aspects of the various problems will
be given special attention. In particular, for infinite Coxeter
groups one may ask for algorithms that are accessible to finite
state automata.

Discover, if possible, connections between the formal language of reduced words in W (a Coxeter group) and topological properties
of geometrical objects associated with the reflection representation
of W.







