- #1

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(see attachment - Kane _ Reflection Groups and Invariant Theory - pages 6-7)

On page 6 Kane mentions he is working in [itex] \ell [/itex] dimensional Euclidean space ie [itex] E = R^{\ell}[/itex] where [itex] R^{\ell}[/itex] has the usual inner product (x,y).

In defining reflections with respect to vectors Kane writes:

" Given [itex] 0 \ne \alpha \in E [/itex] let [itex] H_{\alpha}\subset E [/itex] be the hyperplane

[itex] H_{\alpha} = \{ x | (x, \alpha ) = 0 \} [/itex]

We then define the reflection [itex] s_{\alpha} : E \longrightarrow E [/itex] by the rules

[itex] s_{\alpha} \cdot x = x [/itex] if [itex] x \in H_{\alpha} [/itex]

[itex] s_{\alpha} \cdot \alpha = - \alpha [/itex] "

Then Kane states that the following two properties follow:

(1) [itex] s_{\alpha} \cdot x = x - [2 ( x, \alpha) / (\alpha, \alpha)] \alpha [/itex] for all [itex] x \in E [/itex]

(2) [itex] s_{\alpha} [/itex] is orthogonal, ie [itex] ( s_{\alpha} \cdot x , s_{\alpha} \cdot y ) = (x,y) [/itex] for all [itex] x, y \in E [/itex]

I would appreciate help to show (1) and (2) above.

Peter