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Pythagoras0 (토론 | 기여) (새 문서: ==메모== * http://www.math.harvard.edu/~kronheim/yaft.pdf * 각각의 $\alpha\in \Delta^+$에 대하여, $$ \begin{aligned} \omega_{\alpha}(\theta^{\vee}) & = \omega_{\alpha}(\sum_{...) |
Pythagoras0 (토론 | 기여) |
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| 1번째 줄: | 1번째 줄: | ||
==메모== | ==메모== | ||
* http://www.math.harvard.edu/~kronheim/yaft.pdf | * http://www.math.harvard.edu/~kronheim/yaft.pdf | ||
| − | * 각각의 | + | * 각각의 <math>\alpha\in \Delta^+</math>에 대하여, |
| − | + | :<math> | |
\begin{aligned} | \begin{aligned} | ||
\omega_{\alpha}(\theta^{\vee}) & = \omega_{\alpha}(\sum_{\beta \in \Delta^+} n_{\beta}^{\vee} \beta^{\vee}) \\ | \omega_{\alpha}(\theta^{\vee}) & = \omega_{\alpha}(\sum_{\beta \in \Delta^+} n_{\beta}^{\vee} \beta^{\vee}) \\ | ||
& = n_{\alpha}^{\vee}, | & = n_{\alpha}^{\vee}, | ||
\end{aligned} | \end{aligned} | ||
| − | + | </math> | |
| − | + | :<math> | |
\begin{aligned} | \begin{aligned} | ||
\langle \omega_{\alpha},\theta \rangle &= \omega_{\alpha}(\theta^{\dagger}) \\ | \langle \omega_{\alpha},\theta \rangle &= \omega_{\alpha}(\theta^{\dagger}) \\ | ||
| 14번째 줄: | 14번째 줄: | ||
& = \frac{\langle \theta,\theta \rangle}{2} n_{\alpha}^{\vee} | & = \frac{\langle \theta,\theta \rangle}{2} n_{\alpha}^{\vee} | ||
\end{aligned} | \end{aligned} | ||
| − | + | </math> | |
* 따라서 | * 따라서 | ||
| − | + | :<math> | |
\begin{aligned} | \begin{aligned} | ||
\langle \rho,\theta \rangle & = \langle \sum_{\alpha\in \Delta^+} \omega_{\alpha},\theta \rangle \\ | \langle \rho,\theta \rangle & = \langle \sum_{\alpha\in \Delta^+} \omega_{\alpha},\theta \rangle \\ | ||
| 22번째 줄: | 22번째 줄: | ||
& = \frac{h^{\vee}-1}{2h^{\vee}} | & = \frac{h^{\vee}-1}{2h^{\vee}} | ||
\end{aligned} | \end{aligned} | ||
| − | + | </math>$ | |
2020년 11월 16일 (월) 04:24 판
메모
- http://www.math.harvard.edu/~kronheim/yaft.pdf
- 각각의 \(\alpha\in \Delta^+\)에 대하여,
\[ \begin{aligned} \omega_{\alpha}(\theta^{\vee}) & = \omega_{\alpha}(\sum_{\beta \in \Delta^+} n_{\beta}^{\vee} \beta^{\vee}) \\ & = n_{\alpha}^{\vee}, \end{aligned} \] \[ \begin{aligned} \langle \omega_{\alpha},\theta \rangle &= \omega_{\alpha}(\theta^{\dagger}) \\ & = \frac{\langle \theta,\theta \rangle}{2}\omega_{\alpha}(\theta^{\vee}) \\ & = \frac{\langle \theta,\theta \rangle}{2} n_{\alpha}^{\vee} \end{aligned} \]
- 따라서
\[ \begin{aligned} \langle \rho,\theta \rangle & = \langle \sum_{\alpha\in \Delta^+} \omega_{\alpha},\theta \rangle \\ & = \sum_{\alpha\in \Delta^+} \frac{\langle \theta,\theta \rangle}{2} n_{\alpha}^{\vee} \\ & = \frac{h^{\vee}-1}{2h^{\vee}} \end{aligned} \]$