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Let $\Phi$ be an irreducible root system and $\Delta$ a simple system (base). Let $W$ be the Weyl group of $\Phi$. Let $\theta$ be the highest root and $h^\vee$ be the dual Coxeter number. Choose the shortest $w \in W$ such that $w(\alpha_i)=\theta$ for some simple root $\alpha_i$. I got the result that the length of $w$ can be determined by $l(w) = h^\vee −2$ (for instance see Lemma 5.1 in here). However I'm not given yet any explicit proof for that result. Can anyone give me proof or references where this formula is introduced and proved?

Thanks in advance.

Let $\Phi$ be an irreducible root system and $\Delta$ a simple system (base). Let $W$ be the Weyl group of $\Phi$. Let $\theta$ be the highest root and $h^\vee$ be the dual Coxeter number. Choose the shortest $w \in W$ such that $w(\alpha_i)=\theta$ for some simple root $\alpha_i$. I got the result that the length of $w$ can be determined by $l(w) = h^\vee −2$. However I'm not given yet any explicit proof for that result. Can anyone give me proof or references where this formula is introduced and proved?

Thanks in advance.

Let $\Phi$ be an irreducible root system and $\Delta$ a simple system (base). Let $W$ be the Weyl group of $\Phi$. Let $\theta$ be the highest root and $h^\vee$ be the dual Coxeter number. Choose the shortest $w \in W$ such that $w(\alpha_i)=\theta$ for some simple root $\alpha_i$. I got the result that the length of $w$ can be determined by $l(w) = h^\vee −2$ (for instance see Lemma 5.1 in here). However I'm not given yet any explicit proof for that result. Can anyone give me proof or references where this formula is introduced and proved?

Thanks in advance.

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Jim Humphreys
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Length of Weyl group element mapping heighesthighest root to a simple root

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Length of Weyl element mapping heighest root to a simple root

Let $\Phi$ be an irreducible root system and $\Delta$ a simple system (base). Let $W$ be the Weyl group of $\Phi$. Let $\theta$ be the highest root and $h^\vee$ be the dual Coxeter number. Choose the shortest $w \in W$ such that $w(\alpha_i)=\theta$ for some simple root $\alpha_i$. I got the result that the length of $w$ can be determined by $l(w) = h^\vee −2$. However I'm not given yet any explicit proof for that result. Can anyone give me proof or references where this formula is introduced and proved?

Thanks in advance.