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 \(F\), which is generated by a finite number of reflections.

There are no restrictions on the field \(F\) and there is no requirement for a reflection to be a transformation of order two. However, if \(F\) 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 ChapGrpCox).

The books [Lehrer and Taylor, 2009], [Broué, 2010] or [Kane, 2001] 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 [Humphreys, 1990].