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

Let $R^+$ be the <positive root>[positive roots], $W$ the <Weyl group>, $\ell(w)$ its <Coxeter length>, $\rho$ the <half-sum of positive roots>, and $\alpha^\vee$ a <coroot>. For a dominant integral highest weight $\lambda$, the <Weyl character formula> is
$$
\operatorname{ch}L_\lambda
=\frac{\sum_{w\in W}(-1)^{\ell(w)}e^{w(\lambda+\rho)}}
{\sum_{w\in W}(-1)^{\ell(w)}e^{w\rho}}
=\frac{\sum_{w\in W}(-1)^{\ell(w)}e^{w(\lambda+\rho)}}
{e^\rho\prod_{\alpha\in R^+}(1-e^{-\alpha})}.
$$
Taking the value at the identity gives the <Weyl dimension formula>
$$
\dim L_\lambda
=\prod_{\alpha\in R^+}
\frac{\langle\lambda+\rho,\alpha^\vee\rangle}
{\langle\rho,\alpha^\vee\rangle}.
$$

For the q-character convention relevant to the <Principal sl2 subalgebra>, set $h_{\mathrm{pr}}=2\rho^\vee$, so $\alpha_i(h_{\mathrm{pr}})=2$ for every simple root, and define
$$
\operatorname{ch}_qL_\lambda
=\sum_\mu(\dim L_\lambda[\mu])q^{\mu(h_{\mathrm{pr}})}.
$$
The q-character formula is the principal specialization of the <Weyl character formula>:
$$
\operatorname{ch}_qL_\lambda
=\frac{\sum_{w\in W}(-1)^{\ell(w)}q^{\langle w(\lambda+\rho),2\rho^\vee\rangle}}
{\sum_{w\in W}(-1)^{\ell(w)}q^{\langle w\rho,2\rho^\vee\rangle}}.
$$

Solved by gpt-5.6-sol high.

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 102 / 4 / i / Solution

{c}
{wiki}

The dimension obtained by evaluating the <Weyl character formula> at the identity is
$$
\dim L_\lambda=\prod_{\alpha\in R^+}
\frac{\langle\lambda+\rho,\alpha^\vee\rangle}
{\langle\rho,\alpha^\vee\rangle}.
$$

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 Incoming links: Weyl dimension formula