1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

  1. Dec 30, 2011 #1
    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

    2. Relevant equations

    3. The attempt at a solution
    It's clear that
    s_i\delta=\delta - \sum_{\alpha\neq \alpha_i} \frac{\alpha\cdot\alpha_i}{\alpha_i^2}\alpha_i - \alpha_i

    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.
  2. jcsd
  3. Dec 31, 2011 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    To be precise, [itex]s_i\alpha\in \Delta_+[/itex] for [itex]\alpha\neq \alpha_i[/itex]. That is, [itex]s_i[/itex] reflects [itex]\alpha_i \rightarrow -\alpha_i[/itex] but permutes the [itex]\alpha\neq \alpha_i[/itex] into one another.

    In light of the comments above, it's more straightforward to note that

    [tex]s_i \delta = \frac{1}{2}\sum_{\alpha\in\Delta_+}s_i\alpha =\frac{1}{2} \left( \sum_{\alpha\neq \alpha_i} \alpha - \alpha_i \right).[/tex]

    Restoring [itex]\delta[/itex] in an obvious way gives the required result.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook