Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 102 / 4 / b / Solution

Use the C2 root system realization in an orthonormal basis ${\epsilon }_1,{\epsilon }_2$ with simple roots ${\alpha }_1={\epsilon }_1-{\epsilon }_2$ and ${\alpha }_2=2{\epsilon }_2$. This numbers the short root first, as required. The fundamental weights and half-sum of positive roots areso $\lambda +\rho =(a+b+2){\epsilon }_1+(b+1){\epsilon }_2$. For the four positive roots, the factors of the Weyl dimension formula areMultiplying yields the Weyl dimension formula for C2:As checks, $V({\omega }_1)$ has dimension $4$, $V({\omega }_2)$ dimension $5$, and $V(2{\omega }_1)$ dimension $10$.