Choose positive roots $\Phi^+$ and let
$$
\rho=\frac12\sum_{\alpha\in\Phi^+}\alpha
$$
be the <Weyl vector>. For a <dominant integral weight> $\lambda$, the <Weyl dimension formula> is
$$
\boxed{\dim V(\lambda)
=\prod_{\alpha\in\Phi^+}
\frac{\langle\lambda+\rho,\alpha^\vee\rangle}
{\langle\rho,\alpha^\vee\rangle}}.
$$
Here $\alpha^\vee$ is the coroot and $\langle\ ,\ \rangle$ is the natural weight-coroot pairing.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 102 / 4 / a / Solution

{wiki=Weight_(representation_theory)#Dominant_integral_weights}

An integral weight $\lambda$ is dominant when $\langle\lambda,\alpha_i^\vee\rangle\geq0$ for every simple coroot. Equivalently, it is a nonnegative integer combination of the fundamental weights.


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