Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 102 / 5 / c / i / Solution

Realize the B2 root system in ${\mathbb{R}}^2$ asChoose the short simple root and long simple rootThenas follows from $⟨{\omega }_i,{\alpha }_j^{\vee }⟩={\delta }_{ij}$. The roots form a square from the long roots with the four short roots on the coordinate axes; the double edge in the Dynkin diagram points toward ${\alpha }_1$.

The positive roots areFor $\lambda =a{\omega }_1+b{\omega }_2$, substituting their coroot pairings in the Weyl dimension formula givesThis is the Weyl dimension formula for B2.