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Lie Groups pp 129-144 | Cite as

The Root System

  • Daniel Bump
Chapter
Part of the Graduate Texts in Mathematics book series (GTM, volume 225)

Abstract

A Euclidean space is a real vector space \(\mathcal{V}\) endowed with an inner product, that is, a positive definite symmetric bilinear form. We denote this inner product by \(\left \langle \;,\;\right \rangle\). If \(0\neq \alpha \in \mathcal{V}\), consider the transformation \(s_{\alpha }: \mathcal{V}\longrightarrow \mathcal{V}\) given by
$$\displaystyle{ s_{\alpha }(x) = x -\frac{2\left \langle x,\alpha \right \rangle } {\left \langle \alpha,\alpha \right \rangle } \alpha. }$$
(18.1)
This is the reflection attached to α. Geometrically, it is the reflection in the plane perpendicular to α. We have \(s_{\alpha }(\alpha ) = -\alpha\), while any element of that plane (with \(\left \langle x,\alpha \right \rangle = 0\)) is unchanged by s α.

References

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    T. Springer. Reductive groups. In Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII, pages 3–27. Amer. Math. Soc., Providence, R.I., 1979.Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Daniel Bump
    • 1
  1. 1.Department of MathematicsStanford UniversityStanfordUSA

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