For a dominant integral weight $\lambda$, the <Weyl character formula> is
$$
\operatorname{ch}L_\lambda=
\frac{\sum_{w\in W}(-1)^{\ell(w)}e^{w(\lambda+\rho)}}
{\sum_{w\in W}(-1)^{\ell(w)}e^{w\rho}}.
$$
Here $W$ is the <Weyl group>, $\ell(w)$ its <Coxeter length>, $\rho$ the <Weyl vector>, and $\operatorname{ch}$ the <formal character of a weight module>. Taking the limit at the identity gives the <Weyl dimension formula>
$$
\dim L_\lambda=
\prod_{\alpha\in R^+}
\frac{\langle\lambda+\rho,\alpha^\vee\rangle}
{\langle\rho,\alpha^\vee\rangle}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 102 / 4 / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-102/4/i/solution