 Root string

A root string is the set $S_{\alpha ,\beta }=\Phi \cap (\beta +\mathbb{Z}\alpha )$ for roots $\alpha ,\beta$ of a root system $\Phi$. For nonproportional roots $\alpha$ and $\beta$ of a reduced crystallographic root system, the roots of the form $\beta +n\alpha$ form an unbroken root string $\beta -p\alpha ,\dots ,\beta +q\alpha$, and $p-q=⟨\beta ,{\alpha }^{\vee }⟩$. The root-string theorem explains the endpoint relation. If $\beta =\pm \alpha$, zero is missing and this consecutive-string statement does not apply.