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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.

Keywords

Root System Positive Root Weyl Group Simple Root Dynkin Diagram 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1972

Authors and Affiliations

  • James E. Humphreys
    • 1
  1. 1.University of MassachusettsAmherstUSA

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