grandes-ecoles 2010 QID1

grandes-ecoles · France · centrale-maths2__pc Groups Group Order and Structure Theorems

In this question, the space $E$ has dimension $n = 2$. For every root system $\mathcal { R }$ of $E$, we set $$\theta _ { \mathcal { R } } = \min \left\{ \theta _ { \alpha , \beta } \mid ( \alpha , \beta ) \in \mathcal { R } ^ { 2 } , \alpha \neq \beta \text { and } \alpha \neq - \beta \right\}$$ Show that $\theta _ { \mathcal { R } }$ is well-defined and equals $\pi / 2 , \pi / 3 , \pi / 4$ or $\pi / 6$.

In this question, the space $E$ has dimension $n = 2$. For every root system $\mathcal { R }$ of $E$, we set
$$\theta _ { \mathcal { R } } = \min \left\{ \theta _ { \alpha , \beta } \mid ( \alpha , \beta ) \in \mathcal { R } ^ { 2 } , \alpha \neq \beta \text { and } \alpha \neq - \beta \right\}$$
Show that $\theta _ { \mathcal { R } }$ is well-defined and equals $\pi / 2 , \pi / 3 , \pi / 4$ or $\pi / 6$.

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