Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 302 / 4 / a / v / Solution

The highest weight of a representation is the weight $\Lambda$ of a nonzero weight vector annihilated by every positive-root operator. Every other weight is obtained from $\Lambda$ by subtracting a nonnegative integer combination of simple roots. A finite-dimensional irreducible representation of a complex semisimple Lie algebra is determined by its dominant integral highest weight.