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Take a finite Cartan datum with index set $I$, simple roots $\{\alpha_i\mid i \in I\}$ and positive roots $\Phi^+$. Let $2\rho=\sum_{\alpha\in\Phi^+}\alpha$ be the sum of the positive roots and write $$2\rho = \sum_{i\in I}n_i\alpha_i.$$You can find tables of these coefficients, for example, in the appendix to Bourbaki, chapters 4-6.

My question: what is the significance of the numbers $n_i$? In particular, do they count anything?

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