= Solution
Number the <G2 root system> with $\alpha_1$ short and $\alpha_2$ long, normalized by $(\alpha_1,\alpha_1)=2$, $(\alpha_2,\alpha_2)=6$ and $(\alpha_1,\alpha_2)=-3$. Its positive roots 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 the negatives complete the twelve-root system. The short roots form one hexagon, and the long roots form a rotated hexagon whose radius is $\sqrt3$ times larger. The picture also supplies the crystal requested in the next part.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/iii/paper-102-g2-roots-crystal.png]
{title=The type-G2 root system and defining crystal}
The <fundamental weights> are $\omega_1=2\alpha_1+\alpha_2$, $\omega_2=3\alpha_1+2\alpha_2$, and $\rho=\omega_1+\omega_2$. The six positive <coroots>, in coordinates relative to $\alpha_1^\vee,\alpha_2^\vee$, are $(1,0),(0,1),(1,3),(2,3),(1,1),(1,2)$. Applying the <Weyl dimension formula> to $\lambda=n_1\omega_1+n_2\omega_2$ therefore gives the <G2 dimension polynomial>
$$
\boxed{\dim L(\lambda)=\frac{(n_1+1)(n_2+1)(n_1+n_2+2)
(n_1+2n_2+3)(n_1+3n_2+4)(2n_1+3n_2+5)}{120}.}
$$
The denominator is the product $1\cdot1\cdot2\cdot3\cdot4\cdot5$ of the corresponding pairings with $\rho$. As checks, the weights $\omega_1,\omega_2,2\omega_1$ give dimensions $7,14,27$ respectively.
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