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

For simple roots ${\alpha }_i$, the Cartan matrix isIts diagonal entries are two, while its off-diagonal entries encode the angles and relative lengths in the Dynkin diagram.

The fundamental weights ${\omega }_i$ are the basis dual to the simple coroots:

The Dynkin labels of a weight $\lambda$ areso $\lambda ={\sum }_i{\lambda }^i{\omega }_i$.

A dominant weight has nonnegative Dynkin labels. A dominant integral weight additionally lies in the weight lattice, so all those labels are nonnegative integers.

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.

Pairing ${\alpha }_i={\sum }_j{c}_{ij}{\omega }_j$ with ${\alpha }_k^{\vee }$ gives $c_{ik}=A_{ik}$. Hence the relation between the two bases is