Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 4 / 3 / i / Solution

Let $\mathfrak{g}$ be a complex semisimple Lie algebra, choose a Cartan subalgebra and a positive root system $R^{+}$, and let $\lambda$ be a dominant integral highest weight. Write $L(\lambda )$ for the corresponding finite-dimensional irreducible representation, ${\alpha }^{\vee }=2\alpha /(\alpha ,\alpha )$ for a coroot, and $\rho =\frac{1}{2}{\sum }_{\alpha \in R^{+}}\alpha$ for the Weyl vector. The Weyl dimension formula isThe bracket denotes the natural pairing of weights with coroots. The normalization of an invariant inner product cancels from each ratio, so the formula can equivalently use roots instead of coroots in both numerator and denominator.