 Root-string theorem

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.