Borovik A., Borovik A. - Mirrors and Reflections - The Geometry of Finite Reflection Groups.pdf

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Mirrors and Reections:
The Geometry of Finite Reection Groups
Incomplete Draft Version 01
Alexandre V. Borovik
alexandre.borovik@umist.ac.uk
Anna S. Borovik
anna.borovik@freenet.co.uk
25 February 2000
A. & A. Borovik Mirrors and Reections Version 01 25.02.00
i
Introduction
This expository text contains an elementary treatment of nite groups gen-
erated by reections. There are many splendid books on this subject, par-
ticularly [H] provides an excellent introduction into the theory. The only
reason why we decided to write another text is that some of the applications
of the theory of reection groups and Coxeter groups are almost entirely
based on very elementary geometric considerations in Coxeter complexes.
The underlying ideas of these proofs can be presented by simple drawings
much better than by a dry verbal exposition. Probably for the reason of
their extreme simplicity these elementary arguments are mentioned in most
books only briey and tangently.
We wish to emphasize the intuitive elementary geometric aspects of
the theory of reection groups. We hope that our approach allows an
easy access of a novice mathematician to the theory of reection groups.
This aspect of the book makes it close to [GB]. We realise, however,
that, since classical Geometry has almost completely disappeared from
the schools' and Universities' curricula, we need to smugle it back and
provide the student reader with a modicum of Euclidean geometry and
theory of convex polyhedra. We do not wish to appeal to the reader's
geometric intuition without trying rst to help him or her to develope
it. In particular, we decided to saturate the book with visual material.
Our sketches and diagrams are very unsophisticated; one reason for this
is that we lack skills and time to make the pictures more intricate and
aesthetically pleasing, another is that the book was tested in a M. Sc.
lecture course at UMIST in Spring 1997, and most pictures, in their even
less sophisticated versions, were rst drawn on the blackboard. There was
no point in drawing pictures which could not be reproduced by students
and reused in their homework. Pictures are not for decoration, they are
indispensable (though maybe greasy and soiled) tools of the trade.
The reader will easily notice that we prefer to work with the mirrors
of reections rather than roots. This approach is well known and fully
exploited in Chapter 5, x 3 of Bourbaki's classical text [Bou]. We have
combined it with Tits' theory of chamber complexes [T] and thus made
the exposition of the theory entirely geometrical.
The book contains a lot of exercises of dierent level of diculty. Some
of them may look irrelevant to the subject of the book and are included for
the sole purpose of developing the geometric intuition of a student. The
more experienced reader may skip most or all exercises.
ii
Prerequisites
Formal prerequisites for reading this book are very modest. We assume
the reader's solid knowledge of Linear Algebra, especially the theory of
orthogonal transformations in real Euclidean spaces. We also assume that
they are familiar with the following basic notions of Group Theory:
groups; the order of a nite group; subgroups; normal sub-
groups and factorgroups; homomorphisms and isomorphisms;
permutations, standard notations for them and rules of their
multiplication; cyclic groups; action of a group on a set.
You can nd this material in any introductory text on the subject. We
highly recommend a splendid book by M. A. Armstrong [A] for the rst
reading.
A. & A. Borovik Mirrors and Reections Version 01 25.02.00
iii
Acknowledgements
The early versions of the text were carefully read by Robert Sandling and
Richard Booth who suggested many corrections and improvements.
Our special thanks are due to the students in the lecture course at
UMIST in 1997 where the rst author tested this book:
Bo Ahn, Ayse Berkman, Richard Booth, Nazia Kalsoom, Vaddna
Nuth.
iv

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