 Weight string

For a finite-dimensional representation of a complex semisimple Lie algebra, the weights on a line $\mu +\mathbb{Z}\alpha$ form a contiguous string from $\mu +p\alpha$ to $\mu -q\alpha$, with $q-p=⟨\mu ,{\alpha }^{\vee }⟩$. Restriction to the associated sl2 subalgebra gives nested strings of the same parity, so the string endpoints determine the weight set but not its multiplicities.