Root-string theorem
= Root-string theorem
For nonproportional roots $\gamma,\delta$ of a reduced crystallographic <root system>, $S_{\gamma,\delta}=\{\delta-p\gamma,\ldots,\delta+q\gamma\}$ is consecutive, and $p-q=\langle\delta,\gamma^\vee\rangle$. 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 $\langle\delta,\gamma^\vee\rangle+2n$; 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 $\pm1$ 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 $1-\langle\delta,\gamma^\vee\rangle$. The nonproportional hypothesis excludes $\delta=\pm\gamma$, where zero is not a root and the consecutive-string statement fails.