My Math Forum Finite Reflection Groups in Two Dimensions - R2

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 April 21st, 2012, 11:01 PM #1 Newbie   Joined: Jan 2012 From: Tasmania, Australia Posts: 14 Thanks: 0 Finite Reflection Groups in Two Dimensions - R2 I am seeking to understand reflection groups and am reading Grove and Benson: Finite Reflection Groups On page 6 (see attachment - pages 5 -6 Grove and Benson) we find the following statement: It is easy to verify (Exercise 2.1) that the vector $x_1= (cos \ \theta /2, sin \ \theta /2 )$ is an eigenvector having eigenvalue 1 for T, so that the line $L= \{ \lambda x_1 : \lambda \in \mathbb{R} \}$ is left pointwise fixed by T. I am struggling to see why it follows that L above is left pointwise fixed by T (whatever that means exactly! - can someone please clarify this matter?). Can someone please help - I am hoping to be able to formally and explicitly justify the statement. The preamble to the above statement is given in the attachment, including the definition of T Notes (see attachment) 1. T belongs to the group of all orthogonal transformatios, $O ( \mathbb{R} )$. 2. Det T = -1 For other details see attachment Peter
April 21st, 2012, 11:03 PM   #2
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Re: Finite Reflection Groups in Two Dimensions - R2

Attached Files
 Grove and Benson - Finite Reflection Groups -Pages 5 -6.pdf (53.4 KB, 8 views)

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