Weight string ($q-p=⟨\mu ,{\alpha }^{\vee }⟩$)

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.

Start with the highest weight and process weights by increasing height below it. For each simple root, compute the upper endpoint from the already known higher weights, use the weight string endpoint relation, and append all downward steps. Repeat until the finite set stabilizes. This recovers the weights of a finite-dimensional Irreducible Lie algebra representation without determining weight multiplicities.