Choose a short simple root $\alpha_1$ and a long simple root $\alpha_2$, with $|\alpha_2|^2=3|\alpha_1|^2$ and angle $150^\circ$. The six positive roots of the <G2 root system> are
$$
\alpha_1,\ \alpha_2,\ \alpha_1+\alpha_2,\ 2\alpha_1+\alpha_2,\ 3\alpha_1+\alpha_2,\ 3\alpha_1+2\alpha_2,
$$
and their negatives complete the two concentric hexagons of short and long roots. The <fundamental weight>[fundamental weights] are
$$
\omega_1=2\alpha_1+\alpha_2,
\qquad
\omega_2=3\alpha_1+2\alpha_2,
$$
so $\omega_1$ is itself a short root and $\omega_2$ is the <highest root>.

The seven-dimensional representation $L_{\omega_1}$ has weight set
$$
0,\quad
\pm\alpha_1,\quad
\pm(\alpha_1+\alpha_2),\quad
\pm(2\alpha_1+\alpha_2),
$$
each with <weight multiplicity> one. Their positive heights are $1,2,3$, so the <q-character of a highest-weight representation> is
$$
\operatorname{ch}_qL_{\omega_1}
=q^6+q^4+q^2+1+q^{-2}+q^{-4}+q^{-6}.
$$
This is one $\mathfrak{sl}_2$ weight string, hence
$$
L_{\omega_1}\downarrow\mathfrak{sl}_2^{\mathrm{pr}}\cong V_6.
$$

The representation $L_{\omega_2}$ is the fourteen-dimensional <Adjoint representation>. Its nonzero weights are the twelve roots, and its zero-weight space is the two-dimensional <Cartan subalgebra>. The positive root heights are $1,1,2,3,4,5$, so
$$
\begin{aligned}
\operatorname{ch}_qL_{\omega_2}
={}&q^{10}+q^8+q^6+q^4+2q^2+2\\
&+2q^{-2}+q^{-4}+q^{-6}+q^{-8}+q^{-10}.
\end{aligned}
$$
Splitting this into ordinary $\mathfrak{sl}_2$ strings gives
$$
L_{\omega_2}\downarrow\mathfrak{sl}_2^{\mathrm{pr}}
\cong V_{10}\oplus V_2.
$$

Solved by gpt-5.6-sol high.

Solution

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

{wiki=Weight_(representation_theory)#Weight_multiplicity}

The multiplicity of a weight $\mu$ is the dimension of its <weight space>.

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 Incoming links: Weight multiplicity