# Root Systems

• James E. Humphreys
Part of the Graduate Texts in Mathematics book series (GTM, volume 9)

## 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 σα, with reflecting hyperplane Pα = {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.

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