(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let L be a simple compact Lie group, and [itex] \Delta_+[/itex] is the set of positive roots. I have previously shown that if [itex]\alpha\in\Delta_+[/itex] and [itex]\alpha_i[/itex] is a simple root, then [itex]s_i\alpha\in \Delta_+[/itex] where s_i is the Weyl reflection associated with [itex]\alpha_i[/itex].

Now, let [itex]\delta = \frac{1}{2}\sum_{\alpha\in\Delta_+}\alpha[/itex]. I want to show that

[tex]

s_i\delta=\delta-\alpha_i

[/tex]

2. Relevant equations

3. The attempt at a solution

It's clear that

[tex]

s_i\delta=\delta - \sum_{\alpha\neq \alpha_i} \frac{\alpha\cdot\alpha_i}{\alpha_i^2}\alpha_i - \alpha_i

[/tex]

But I have no idea how to show that [itex]\alpha\cdot\alpha_i=0\quad \forall\alpha\neq\alpha_i[/itex]. I cannot make appeal to the fact that delta might be a sum of fundamental weights because that's what I need to show later on.

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# Homework Help: Showing that half-sum of positive roots is the sum of fundamental weights

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