## Abstract

Throughout this chapter we are concerned with a fixed euclidean space E, i.e., a finite dimensional vector space over **R** endowed with a positive definite symmetric bilinear form (α,*β*). Geometrically, a **reflection** in E is an invertible linear transformation leaving pointwise fixed some **hyperplane** (subspace of codimension one) and sending any vector orthogonal to that **hyperplane** into its negative. Evidently a reflection is *orthogonal*, i.e., preserves the inner product on E. Any nonzero vector α determines a reflection *σ*_{α}, with **reflecting hyperplane** *P*_{α} = {*β* є E|(*β*, α) = 0}. Of course, nonzero vectors proportional to a yield the same reflection. It is easy to write down an explicit formula: \({\sigma _\alpha }(\beta ) = \beta - \frac{{2(\beta ,\alpha )}}{{(\alpha ,\alpha )}}\alpha \) (This works, because it sends α to — α and fixes all points in *P*_{ α }.) Since the number 2(*β*, α)/(α, α) occurs frequently, we abbreviate it by <*β*, α>. Notice that <*β*, α> is linear only in the first variable.

## Keywords

Root System Positive Root Weyl Group Simple Root Dynkin Diagram## Preview

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