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.

Solution

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

{wiki=Root_system#Positive_roots_and_simple_roots}

A choice of positive roots selects exactly one of $\alpha$ and $-\alpha$ and is closed under addition whenever the sum is a root.


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