= Solution
The <G2 dimension polynomial> gives dimension seven at the short-root <fundamental weight>, so
$$
\boxed{V=L(\omega_1),\qquad\omega_1=2\alpha_1+\alpha_2.}
$$
The <crystal of the seven-dimensional G2 representation> is a chain whose weights, in order, are
$$
2\alpha_1+\alpha_2,\quad\alpha_1+\alpha_2,\quad\alpha_1,\quad0,
\quad-\alpha_1,\quad-\alpha_1-\alpha_2,\quad-2\alpha_1-\alpha_2.
$$
Its successive arrow colors are $1,2,1,1,2,1$, as drawn above. Each arrow subtracts its indicated <simple root>; in particular the central three vertices form a length-two string of color one. The six nonzero weights are exactly the six short roots, each with <multiplicity> one, and the zero weight also has <multiplicity> one.
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