.. _SectGrpRfllIntro:

.. _introduction:

Introduction
============

A reflection is a diagonalisable linear transformation of finite order whose
space of fixed points is a hyperplane. A reflection group is a finite
dimensional linear group over a field :math:`F`, which is generated by a finite
number of reflections.

There are no restrictions on the field :math:`F` and there is no requirement for
a reflection to be a transformation of order two. However, if :math:`F` is a
real field, every reflection does have order two and there is a much richer
theory. In particular, every Coxeter group is a real reflection group (see
Chapter :ref:`ChapGrpCox`).

The books :cite:`LehrerTaylor`, :cite:`Broue` or
:cite:`KaneRefl` are useful references for complex reflection
groups. Standard references for the theory of real reflection groups include
:raw-latex:`\cite[Chapters 4, 5, 6]{BourbakiLie}` and
:cite:`HumphreysRefl`.