Root string ($S_{\alpha ,\beta }=\Phi \cap (\beta +\mathbb{Z}\alpha )$)

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.

For nonproportional roots $\gamma ,\delta$ of a reduced crystallographic root system, $S_{\gamma ,\delta }=\{\delta -p\gamma ,\dots ,\delta +q\gamma \}$ is consecutive, and $p-q=⟨\delta ,{\gamma }^{\vee }⟩$. Realize the crystallographic root system as the roots of a complex semisimple Lie algebra. The sum of the one-dimensional root spaces in this string is a module for the sl2 subalgebra associated with a root. Its weights are $⟨\delta ,{\gamma }^{\vee }⟩+2n$; they have one fixed parity. Complete reducibility gives a sum of irreducible representations of the sl2 Lie algebra. Two summands of this parity would overlap at weight zero when the parity is even, or at weights $\pm 1$ when it is odd, contradicting the one-dimensional root spaces. Thus there is one irreducible string, whose opposite endpoint weights sum to zero. This proves the relation. For distinct simple roots, $p=0$ and the cardinality is $1-⟨\delta ,{\gamma }^{\vee }⟩$. The nonproportional hypothesis excludes $\delta =\pm \gamma$, where zero is not a root and the consecutive-string statement fails.