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Following Humphreys' Lie algebra, let $\Phi$ be a root system in euclidean space $E=\mathbb{R}^n$.

For every root $\alpha\in\Phi$ let $P_{\alpha}$ be the hyperplane orthogonal to $\alpha$.

Then $E-\cup_{\alpha} P_{\alpha}$ is a (finite) union of open connected subsets of $E$, called Weyl chambers.

Let $\gamma,\gamma'\in E -\cup_{\alpha}P_{\alpha}$ lie in same connected component.

Q. What is algebraic formulation of this statement in terms of $\gamma,\gamma'$, $\alpha$'s and the inner product?

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This means that for all roots $\alpha \in \Phi$,$\langle \gamma, \alpha \rangle$ and $\langle \gamma', \alpha \rangle$ have the same sign. Of course you can restrict to positive roots $\alpha \in \Phi^+$ (for some choice of simple roots).

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